FUZZY MULTI-CRITERIA DECISION MAKING

Springer Optimization and Its Applications VOLUME 16 Managing Editor Panos M. Pardalos (University of Florida) Editor—Combinatorial Opt imization Ding-Zhu Du (University of Texas at Dallas) Advisory Board J. Birge (U niversity of Chicago) C.A. Floudas (Princeton University) F. Giannessi (Universi ty of Pisa) H.D. Sherali (Virginia Polytechnic and State University) T. Terlaky (McMaster University) Y. Ye (Stanford University) Aims and Scope Optimization ha s been expanding in all directions at an astonishing rate during the last few de cades. New algorithmic and theoretical techniques have been developed, the diffu sion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis o n the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics and other sciences. The series Optimization and Its Applications publishes undergraduate a nd graduate textbooks, monographs and state-of-the-art expository works that foc us on algorithms for solving optimization problems and also study applications i nvolving such problems. Some of the topics covered include nonlinear optimizatio n (convex and nonconvex), network flow problems, stochastic optimization, optima l control, discrete optimization, multiobjective programming, description of sof tware packages, approximation techniques and heuristic approaches.

FUZZY MULTI-CRITERIA DECISION MAKING Theory and Applications with Recent Develop ments Edited By CENGIZ KAHRAMAN Istanbul Technical University, Istanbul, Turkey

Cengiz Kahraman Department of Industrial Engineering Istanbul Technical Universi ty Campus Macka 34367 Istanbul Turkey kahramanc@itu.edu.tr ISSN: 1931-6828 ISBN: 978-0-387-76812-0 DOI: 10.1007/978-0-387-76813-7 e-ISBN: 978-0-387-76813-7 Library of Congress Control Number: 2008922672 Mathematics Subject Classificatio n: 03E72 Fuzzy set theory, 03E75 Applications of set theory 2008 Springer Scienc e+Business Media, LLC All rights reserved. This work may not be translated or co pied in whole or in part without the written permission of the publisher (Spring er Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), exc ept for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adapta tion, computer software, or by similar or dissimilar methodology now known or he reafter developed is forbidden. The use in this publication of trade names, trad emarks, service marks, and similar terms, even if they are not identified as suc h, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com

PREFACE Multiple criteria decision making (MCDM) is a modeling and methodological tool f or dealing with complex engineering problems. Decision makers face many problems with incomplete and vague information in MCDM problems since the characteristic s of these problems often require this kind of information. Fuzzy set approaches are suitable to use when the modeling of human knowledge is necessary and when human evaluations are needed. Fuzzy set theory is recognized as an important pro blem modeling and solution technique. Fuzzy set theory has been studied extensiv ely over the past 40 years. Most of the early interest in fuzzy set theory perta ined to representing uncertainty in human cognitive processes. Fuzzy set theory is now applied to problems in engineering, business, medical and related health sciences, and the natural sciences. Over the years there have been successful ap plications and implementations of fuzzy set theory in MCDM. MCDM is one of the b ranches in which fuzzy set theory found a wide application area. Many curriculum s of undergraduate and graduate programs include many courses teaching how to us e fuzzy sets when you face incomplete and vague information. One of these course s is fuzzy MCDM and its applications. This book presents examples of application s of fuzzy sets in MCDM. It contains 22 original research and application chapte rs from different perspectives; and covers different areas of fuzzy MCDM. The bo ok contains chapters on the two major areas of MCDM to which fuzzy set theory co ntributes. These areas are fuzzy multiple-attribute decision making (MADM) and f uzzy multiple-objective decision making (MODM). MADM approaches can be viewed as alternative methods for combining the information in a problem’s decision matrix together with additional information from the decision maker to determine a fina l ranking, screening, or selection from among the alternatives. MODM is a powerf ul tool to assist in the process of searching for decisions that best satisfy a multitude of conflicting objectives. v

vi Preface The classification, review and analysis of fuzzy multi-criteria decisionmaking m ethods are summarized in the first two chapters. While the first chapter classif ies the multi-criteria methods in a general sense, the second chapter focuses on intelligent fuzzy multi-criteria decision making. The rest of the book is divid ed into two main parts. The first part includes chapters on frequently used MADM techniques under fuzziness, e.g., fuzzy Analytic Hierarchy Process (AHP), fuzzy TOPSIS, fuzzy outranking methods, fuzzy weighting methods, and a few applicatio n chapters of these techniques. The third chapter includes the most frequently u sed fuzzy AHP methods and their numerical and didactic examples. The fourth chap ter shows how a fuzzy AHP method can be jointly used with another technique. The fifth chapter summarizes fuzzy outranking methods, which dichotomize preferred alternatives and nonpreferred ones by establishing outranking relationships. The sixth chapter presents another commonly used multi-attribute method, fuzzy TOPS IS and its application to selection among industrial robotic systems. The sevent h chapter includes many fuzzy scoring methods and their applications. The rest o f this part includes the other most frequently used fuzzy MADM techniques in the literature: fuzzy information axiom approach, intelligent fuzzy MADM approaches , gray-related analysis, and neuro-fuzzy approximation. The second part of the b ook includes chapters on MODM techniques under fuzziness, e.g., fuzzy multi-obje ctive linear programming, quasiconcave and non-concave fuzzy multi-objective pro gramming, interactive fuzzy stochastic linear programming, fuzzy multi-objective integer goal programming, gray fuzzy multi-objective optimization, fuzzy multio bjective geometric programming and some applications of these techniques. These methods are the most frequently used MODM techniques in the fuzzy literature. Th e presented methods in this book have been prepared by the authors who are the d evelopers of these techniques. I hope that this book will provide a useful resou rce of ideas, techniques, and methods for additional research on the application s of fuzzy sets in MCDM. I am grateful to the referees whose valuable and highly appreciated works contributed to select the high quality of chapters published in this book. I am also grateful to my research assistant, Dr. Ihsan Kaya, for h is invaluable effort to edit this book. Cengiz Kahraman Istanbul Technical Unive rsity May 2008

CONTENTS Preface......................................................................... ................................ v Contributors................................. ............................................................... xi Multi-Criteri a Decision Making Methods and Fuzzy Sets.......................... 1 Cengiz Kahr aman Intelligent Fuzzy Multi-Criteria Decision Making: Review and Analysis...... ................................................................................ ............... 19 Waiel F. Abd El-Wahed Part I: FUZZY MADM METHODS AND APPLICATIONS Fuzzy Analytic Hierarchy Process and Its Application............................ . 53 Tufan Demirel, Nihan Çetin Demirel, and Cengiz Kahraman A SWOT-AHP Applicatio n Using Fuzzy Concept: E-Government in Turkey .................................. ................................................................. 85 Cengiz Kahr aman, Nihan Çetin Demirel, Tufan Demirel, and Nüfer Yasin Ate Fuzzy Outranking Metho ds: Recent Developments................................ 119 Ahmed Bufardi, Razva n Gheorghe, and Paul Xirouchakis Fuzzy Multi-Criteria Evaluation of Industrial R obotic Systems Using TOPSIS .................................................... .................................... 159 Cengiz Kahraman, Ihsan Kaya, Sezi Çevik, Nüfer Yasin Ates, and Murat Gülbay Fuzzy Multi-Attribute Scoring Methods with Applic ations.................... 187 Cengiz Kahraman, Semra Birgün, and Vedat Zeki Yenen vii

viii Contents Fuzzy Multi-Attribute Decision Making Using an Information Axiom-Based Approach .......................................................................... 209 C engiz Kahraman and Osman Kulak Measurement of Level-of-Satisfaction of Decision Maker in Intelligent Fuzzy-MCDM Theory: A Generalized Approach ................. 235 Pandian Vasant, Arijit Bhattacharya, and Ajith Abraham FMS Selection Under Disparate Level-of-Satisfaction of Decision Making Using an Intelligent Fuzzy-MC DM Model ................ 263 Arijit Bhattacharya, Ajith Abraham, and Pandian Va sant Simulation Support to Grey-Related Analysis: Data Mining Simulation ....... ................................................................................ ........ 281 David L. Olson and Desheng Wu Neuro-Fuzzy Approximation of Multi-Cr iteria Decision-Making QFD Methodology ......................................... ......................................... 301 Ajith Abraham, Pandian Vasant, and Arijit Bhattacharya Part II: FUZZY MODM METHODS AND APPLICATIONS Fuzzy Multiple Objective Linear Programming .................................... . 325 Cengiz Kahraman and Ihsan Kaya Quasi-Concave and Nonconcave FMODM Problems ............................ 339 Chian-Son Yu and Han-Lin Li Interactive Fuzzy Multi-Objective Stochastic Linear Programming....... 375 Masatoshi Sakawa and Ko suke Kato An Interactive Algorithm for Decomposing: The Parametric Space in Fuzz y Multi-Objective Dynamic Programming Problems.................. 409 Mahmoud A. Abo-Sinna, A.H. Amer, and Hend H. EL Sayed Goal Programming Approaches for Solvi ng Fuzzy Integer Multi-criteria Decision-Making Problems........................ .......................431 Omar M. Saad

Contents ix Grey Fuzzy Multi-Objective Optimization ........................................ .....453 P.P. Mujumdar and Subhankar Karmakar Fuzzy Multi-Objective Decision-Mak ing Models and Approaches........483 Jie Lu, Guangquan Zhang, and Da Ruan Fuzzy Optimization via Multi-Objective Evolutionary Computation for Chocolate Manufact uring ............................................523 Fernando Jiménez, Gracia Sánch ez, Pandian Vasant, and José Luis Verdegay Multi-Objective Geometric Programming a nd Its Application in an Inventory Model ....................................... ......................................539 Tapan Kumar Roy Fuzzy Geometric Progra mming with Numerical Examples....................567 Tapan Kumar Roy Index...... ................................................................................ ..................589

CONTRIBUTORS Waiel F. Abd El-Wahed OR & DS Dept., Faculty of Computers & Information, Menoufi a University, Shiben El-Kom, Egypt waeilf@yahoo.com Mahmoud A. Abo-Sinna Departm ent of Basic Engineering Science, Faculty of Engineering, EL-Menoufia University , Shebin EL-kom, P.O. Box 398, 31111 Tanta, AL-Gharbia, Egypt Mabosinna2000@Yaho o.com Ajith Abraham Norwegian Center of Excellence, Center of Excellence for Qua ntifiable Quality of Service, Norwegian University of Science and Technology O.S . Bragstads plass 2E, NO-7491 Trondheim, Norway ajith.abraham@ieee.org Azza H. A mer Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egy pt Nüfer Yasin Ate Istanbul Technical University, Department of Industrial Enginee ring, Besiktas-Istanbul, Turkey yasinn@itu.edu.tr Arijit Bhattacharya Embark Ini tiative Post-Doctoral Research Fellow, School of Mechanical & Manufacturing Engi neering, Dublin City University, Glasnevin, Dublin 9, Ireland arijit.bhattachary a2005@gmail.com xi

xii Contributors Semra Birgün Istanbul Commerce University, Department of Industrial Engineering, Üsküd ar, stanbul sbirgun@iticu.edu.tr Ahmed Bufardi Institute of Production and Robot ics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland Ahmed.Bufardi@eawa g.ch Sezi Çevik Istanbul Technical University, Department of Industrial Engineerin g, Besiktas-Istanbul, Turkey cevikse@itu.edu.tr Nihan Çetin Demirel Yildiz Technic al University, Department of Industrial Engineering, Yildiz-Istanbul, Turkey nih an@yildiz.edu.tr Tufan Demirel Yildiz Technical University, Department of Indust rial Engineering, Yildiz-Istanbul, Turkey tdemirel@yildiz.edu.tr Hend H. El Saye d Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt Razvan Gheorghe Institute of Production and Robotics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland razvan-alex.gheorghe@a3.epfl.ch Murat Gülbay Istan bul Technical University, Department of Industrial Engineering, Besiktas-Istanbu l, Turkey gulbaym@itu.edu.tr

Contributors xiii Fernando Jiménez Dept. Ingeniería de la Información y las Comunicaciones, University o f Murcia, Spain fernan@dif.um.es Cengiz Kahraman Istanbul Technical University, Department of Industrial Engineering, Besiktas-Istanbul, Turkey kahramanc@itu.ed u.tr Subhankar Karmakar Department of Civil Engineering, Indian Institute of Sci ence, Bangalore, India skar@civil.iisc.ernet.in Kosuke Kato Department of Artifi cial Complex Systems Engineering, Graduate School of Engineering, Hiroshima Univ ersity, Japan kato@msl.sys.hiroshima-u.ac.jp Ihsan Kaya Istanbul Technical Unive rsity, Department of Industrial Engineering, Besiktas-Istanbul, Turkey kayai@itu .edu.tr Osman Kulak Pamukkale University, Industrial Engineering Department, Den izli, Turkey okulak@pamukkale.edu.tr Han-Lin Li Institute of Information Managem ent, National Chiao Tung University, Hsinchi 30050, Taiwan hlli@cc.nctu.edu.tw J ie Lu Faculty of Information Technology, University of Technology, Sydney, PO Bo x 123, Broadway, NSW 2007, Australia jielu@it.uts.edu.au

xiv Contributors P.P. Mujumdar Department of Civil Engineering, Indian Institute of Science, Bang alore, India radeep@civil.iisc.ernet.in David L. Olson Department of Management, University of Nebraska, Lincoln, NE 68588-0491, USA dolson3@unl.edu Tapan Kumar Roy Department of Mathematics, Bengal Engineering and Science University, Shibp ur Howrah – 711103, West Bengal, India roy_t_k@yahoo.co.in Da Ruan Belgian Nuclear Research Centre (SCK•CEN) Boeretang 200, 2400 Mol, Belgium druan@sckcen.be Omar M . Saad Department of Mathematics, College of Science, Qatar University, P.O. Box 2713, Doha, Qatar omarr_saad@yahoo.com Masatoshi Sakawa Department of Artificia l Complex Systems Engineering, Graduate School of Engineering, Hiroshima Univers ity, Japan sakawa@msl.sys.hiroshima-u.ac.jp Gracia Sánchez Dept. Ingeniería de la In formación y las Comunicaciones, University of Murcia, Spain gracia@dif.um.es Pandi an Vasant EEE Program Research Lecturer, Universiti Teknologi Petronas, 31750 Tr onoh, BSI, Perak DR, Malaysia pvasant@gmail.com

Contributors xv José Luis Verdegay Dept. Ciencias de la Computaci´on e Inteligencia Artificial, Univ ersity of Granada, Spain verdegay@decsai.ugr.es Desheng Wu RiskLab, University o f Toronto, 1 Spadina Crescent Room, 205, Toronto, Ontario, M5S 3G3 Canada dash@u stc.edu Paul Xirouchakis Institute of Production and Robotics, Ecole Polytechniq ue Fédérale de Lausanne (EPFL), Switzerland Ahmed.Bufardi@eawag.ch Vedat Zeki Yenen Istanbul Commerce University, Department of Industrial Engineering, Üsküdar, stanbul vzyenen@iticu.edu.tr Chian-Son Yu Department of Information Management, Shih Ch ien University, Taipei 10497, Taiwan csyu@mail.usc.edu.tw Guangquan Zhang Facult y of Information Technology, University of Technology, Sydney, PO Box 123, Broad way, NSW 2007, Australia zhangg@it.uts.edu.au

MULTI-CRITERIA DECISION MAKING METHODS AND FUZZY SETS Cengiz Kahraman Department of Industrial Engineering, Istanbul Technical University, 34367 Maçka s tanbul Turkey Abstract: Multi-criteria decision making (MCDM) is one of the well-known topics of decisio n making. Fuzzy logic provides a useful way to approach a MCDM problem. Very oft en in MCDM problems, data are imprecise and fuzzy. In a real-world decision situ ation, the application of the classic MCDM method may face serious practical con straints, because of the criteria containing imprecision or vagueness inherent i n the information. For these cases, fuzzy multi-attribute decision making (MADM) and fuzzy multi-objective decision making (MODM) methods have been developed. I n this chapter, crisp MADM and MODM methods are first summarized briefly and the n the diffusion of the fuzzy set theory into these methods is explained. Some ex amples of recently published papers on fuzzy MADM and MODM are given. Multi-crit eria, multi-attribute, multi-objective, decision making, fuzzy sets Key words: 1. INTRODUCTION In the literature, there are two basic approaches to multiple criteria decision making (MCDM) problems: multiple attribute decision making (MADM) and multiple o bjective decision making (MODM). MADM problems are distinguished from MODM probl ems, which involve the design of a “best” alternative by considering the tradeoffs w ithin a set of interacting design constraints. MADM refers to making selections among some courses of action in the presence of multiple, usually conflicting, a ttributes. In MODM problems, the number of alternatives is effectively C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 1

2 C. Kahraman infinite, and the tradeoffs among design criteria are typically described by con tinuous functions. MADM is the most well-known branch of decision making. It is a branch of a general class of operations research models that deal with decisio n problems under the presence of a number of decision criteria. The MADM approac h requires that the choice (selection) be made among decision alternatives descr ibed by their attributes. MADM problems are assumed to have a predetermined, lim ited number of decision alternatives. Solving a MADM problem involves sorting an d ranking. MADM approaches can be viewed as alternative methods for combining th e information in a problem’s decision matrix together with additional information from the decision maker to determine a final ranking, screening, or selection fr om among the alternatives. Besides the information contained in the decision mat rix, all but the simplest MADM techniques require additional information from th e decision maker to arrive at a final ranking, screening, or selection. In the M ODM approach, contrary to the MADM approach, the decision alternatives are not g iven. Instead, MODM provides a mathematical framework for designing a set of dec ision alternatives. Each alternative, once identified, is judged by how close it satisfies an objective or multiple objectives. In the MODM approach, the number of potential decision alternatives may be large. Solving a MODM problem involve s selection. It has been widely recognized that most decisions made in the real world take place in an environment in which the goals and constraints, because o f their complexity, are not known precisely, and thus, the problem cannot be exa ctly defined or precisely represented in a crisp value (Bellman and Zadeh, 1970) . To deal with the kind of qualitative, imprecise information or even ill-struct ured decision problems, Zadeh (1965) suggested employing the fuzzy set theory as a modeling tool for complex systems that can be controlled by humans but are ha rd to define exactly. Fuzzy logic is a branch of mathematics that allows a compu ter to model the real world the same way that people do. It provides a simple wa y to reason with vague, ambiguous, and imprecise input or knowledge. In Boolean logic, every statement is true or false; i.e., it has a truth value 1 or 0. Bool ean sets impose rigid membership requirements. In contrast, fuzzy sets have more flexible membership requirements that allow for partial membership in a set. Ev erything is a matter of degree, and exact reasoning is viewed as a limiting case of approximate reasoning. Hence, Boolean logic is a subset of Fuzzy logic. Huma n beings are involved in the decision analysis since decision making should take into account human subjectivity,

MCDM Methods and Fuzzy Sets 3 rather than employing only objective probability measures. This makes fuzzy deci sion making necessary. This chapter aims at classifying MADM and MODM methods an d at explaining how the fuzzy sets have diffused into the MCDM methods. 2. MULTI-ATTRIBUTE DECISION MAKING: A CLASSIFICATION OF METHODS MADM methods can be classified as to whether if they are noncompensatory or comp ensatory. The decision maker may be of the view that high performance relative t o one attribute can at least partially compensate for low performance relative t o another attribute, particularly if an initial screening analysis has eliminate d alternatives that fail to meet any minimum performance requirements. Methods t hat incorporate tradeoffs between high and low performance into the analysis are termed “compensatory.” Those methods that do not are termed “noncompensatory.” In their book, Hwang and Yoon (1981) give 14 MADM methods. These methods are explained b riefly below. Additionally five more methods are listed below. 2.1 Dominance An alternative is “dominated” if another alternative outperforms it with respect to at least one attribute and performs equally with respect to the remainder of att ributes. With the dominance method, alternatives are screened such that all domi nated alternatives are discarded. The screening power of this method tends to de crease as the number of independent attributes becomes larger. 2.2 Maximin The principle underlying the maximin method is that “a chain is only as strong as its weakest link.” Effectively, the method gives each alternative a score equal to the strength of its weakest link, where the “links” are the attributes. Thus, it re quires that performance with respect to all attributes be measured in commensura te units (very rare for MADM problems) or else be normalized prior to performing the method.

4 C. Kahraman 2.3 Maximax The viewpoint underlying the maximax method is one that assigns total importance to the attribute with respect to which each alternative performs best. Extendin g the “chain” analogy used in describing the maximin method, maximax performs as if one was comparing alternative chains in search of the best link. The score of ea ch chain (alternative) is equal to the performance of its strongest link (attrib ute). Like the maximin method, maximax requires that all attributes be commensur ate or else prenormalized. 2.4 Conjunctive (Satisficing) The conjunctive method is purely a screening method. The requirement embodied by the conjunctive screening approach is that to be acceptable, an alternative mus t exceed given performance thresholds for all attributes. The attributes (and th us the thresholds) need not be measured in commensurate units. 2.5 Disjunctive The disjunctive method is also purely a screening method. It is the complement o f the conjunctive method, substituting “or” in place of “and.” That is, to pass the disj unctive screening test, an alternative must exceed the given performance thresho ld for at least one attribute. Like the conjunctive method, the disjunctive meth od does not require attributes to be measured in commensurate units. 2.6 Lexicographic The best-known application of the lexicographic method is, as its name implies, alphabetical ordering such as is found in dictionaries. Using this method, attri butes are rank-ordered in terms of importance. The alternative with the best per formance on the most important attribute is chosen. If there are ties with respe ct to this attribute, the next most important attribute is considered, and so on . Note two important ways in which MADM problems typically differ from alphabeti zing dictionary words. First, there are many fewer alternatives in a MADM proble m than words in the dictionary. Second, when the decision matrix contains quanti tative attribute

MCDM Methods and Fuzzy Sets 5 values, there are effectively an infinite number [rather than 26 (i.e., A-Z)] of possible scores with a correspondingly lower probability of ties. 2.7 Lexicographic Semi-Order This is a slight variation on the lexicographic method, where “near-ties” are allowe d to count as ties without any penalty to the alternative, which scores slightly lower within the tolerance (“tie”) window. Counting nearties as ties makes the lexi cographic method less of a “knife-edged” ranking method and more appropriate for MAD M problems with quantitative data in the decision matrix. However, the method ca n lead to intransitive results, wherein A is preferred to B, B is preferred to C , but C is preferred to A. 2.8 Elimination by Aspects This method is a formalization of the well-known heuristic, “process of eliminatio n.” Like the lexicographic method, this evaluation proceeds one attribute at a tim e, starting with attributes determined to be most important. Then, like the conj unctive method, alternatives not exceeding minimum performance requirements—with r espect to the single attribute of interest, in this case—are eliminated. The proce ss generally proceeds until one alternative remains, although adjustment of the performance threshold may be required in some cases to achieve a unique solution . 2.9 Linear Assignment Method This method requires, in addition to the decision matrix data, cardinal importan ce weights for each attribute and rankings of the alternatives with respect to e ach attribute. These information requirements are intermediate between those of the eight methods described previously, and the five methods that follow, in tha t they require ordinal (but not cardinal) preference rankings of the alternative s with respect to each attribute. The primary use of the additional information is to enable compensatory rather than noncompensatory analysis, that is, allowin g good performance on one attribute to compensate for low performance on another . Note at this point that quantitative attribute values (data in the decision ma trix) do not constitute cardinal preference rankings. Attribute values are gener ally noncommensurate across attributes, preference is not necessarily linearly i ncreasing with attribute values, and preference for attribute values

6 C. Kahraman of zero is not generally zero. However, as long as the decision maker can specif y an ordinal correspondence between attribute values and preference, such as “more is better” or “less is better” for each attribute, then the ordinal alternative ranki ngs with respect to each attribute that are needed by the linear assignment meth od are specified uniquely. Thus, the evaluation/performance rankings required by the linear assignment method are easier to derive than the evaluation/performan ce ratings required by the five methods that follow. The cost of using ordinal r ankings rather than cardinal ratings is that the method is only “semi-compensatory ,” in that incremental changes in the performance of an alternative will not enter into the analysis unless the changes are large enough to alter the rank order o f the alternatives. 2.10 Additive Weighting The score of an alternative is equal to the weighted sum of its cardinal evaluat ion/preference ratings, where the weights are the importance weights associated with each attribute. The resulting cardinal scores for each alternative can be u sed to rank, screen, or choose an alternative. The analytical hierarchy process (AHP) is a particular approach to the additive weighting method. 2.11 Weighted Product The weighted product is similar to the additive weighting method. However, inste ad of calculating “sub-scores” by multiplying performance scores times attribute imp ortances, performance scores are raised to the power of the attribute importance weight. Then, rather than summing the resulting subscores across attributes to yield the total score for the alternative, the product of the scores yields the final alternative scores. The weighted product method tends to penalize poor per formance on one attribute more heavily than does the additive weighting method. 2.12 Nontraditional Capital Investment Criteria This method entails pairwise comparisons of the performance gains (over a baseli ne alternative) among attributes, for a given alternative. One attribute must be measured in monetary units. These comparisons are combined to estimate the (mon etary) value attributed to each performance gain, and these values are summed to yield the overall implied value of each

MCDM Methods and Fuzzy Sets 7 alternative. These implied values can be used to select an alternative, to rank alternatives, or presumably to screen alternatives as well. 2.13 TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) The principle behind TOPSIS is simple: The chosen alternative should be as close to the ideal solution as possible and as far from the negative-ideal solution a s possible. The ideal solution is formed as a composite of the best performance values exhibited (in the decision matrix) by any alternative for each attribute. The negative-ideal solution is the composite of the worst performance values. P roximity to each of these performance poles is measured in the Euclidean sense ( e.g., square root of the sum of the squared distances along each axis in the “attr ibute space”), with optional weighting of each attribute. 2.14 Distance from Target This method and its results are also straightforward to describe graphically. Fi rst, target values for each attribute are chosen, which need not be exhibited by any available alternative. Then, the alternative with the shortest distance (ag ain in the Euclidean sense) to this target point in “attribute space” is selected. A gain, weighting of attributes is possible. Distance scores can be used to screen , rank, or select a preferred alternative. 2.15 Analytic Hierarchy Process (AHP) The analytical hierarchy process was developed primarily by Saaty (1980). AHP is a type of additive weighting method. It has been widely reviewed and applied in the literature, and its use is supported by several commercially available, use r-friendly software packages. Decision makers often find it difficult to accurat ely determine cardinal importance weights for a set of attributes simultaneously . As the number of attributes increases, better results are obtained when the pr oblem is converted to one of making a series of pairwise comparisons. AHP formal izes the conversion of the attribute weighting problem into the more tractable p roblem of making a series of pairwise comparisons among competing attributes. AH P summarizes the results of pairwise comparisons in a “matrix of pairwise comparis ons.” For each pair of attributes, the decision

8 C. Kahraman maker specifies a judgment about “how much more important one attribute is than th e other.” Each pairwise comparison requires the decision maker to provide an answe r to the question: “Attribute A is how much more important than Attribute B, relat ive to the overall objective?” 2.16 Outranking Methods (ELECTRE, PROMETHEE, ORESTE) The basic concept of the ELECTRE (ELimination Et Choix Traduisant la Réalité or Elim ination and Choice Translating Reality) method is how to deal with outranking re lation by using pairwise comparisons among alternatives under each criteria sepa rately. The outranking relationship of Aj, describes that even though two two al ternatives, denoted as Ai alternatives i and j do not dominate each other mathem atically, the decision maker accepts the risk of regarding Ai as almost surely b etter than Aj. An alternative is dominated if another alternative outranks it at least in one criterion and equals it in the remaining criteria. The ELECTRE met hod consists of a pairwise comparison of alternatives based on the degree to whi ch evaluation of the alternatives and preference weight confirms or contradicts the pairwise dominance relationship between the alternatives. The decision maker may declare that she/he has a strong, weak, or indifferent preference or may ev en be unable to express his or her preference between two compared alternatives. The other two members of outranking methods are PROMETHEE and ORESTE. 2.17 Multiple Attribute Utility Models Utility theory describes the selection of a satisfactory solution as the maximiz ation of satisfaction derived from its selection. The best alternative is the on e that maximizes utility for the decision maker’s stated preference structure. Uti lity models are of two types additive and multiplicative utility models. The mai n steps in using a multi-attribute utility model can be counted as 1) determinat ion of utility functions for individual attributes, 2) determination of weightin g or scaling factors, 3) determination of the type of utility model, 4) the meas urement of the utility values for each alternative with respect to the considere d attributes, and 5) the selection of the best alternative.

MCDM Methods and Fuzzy Sets 9 2.18 Analytic Network Process In some practical decision problems, it seems to be the case where the local wei ghts of criteria are different for each alternative. AHP has a difficulty in tre ating in such a case since AHP uses the same local weights of criteria for each alternative. To overcome this difficulty, Saaty (1996) proposed the analytic net work process (ANP). ANP permits the use of different weights of criteria for alt ernatives. 2.19 Data Envelopment Analysis Data envelopment analysis (DEA) is a nonparametric method of measuring the effic iency of a decision making unit such as a firm or a public-sector agency, which was first introduced into the operations research literature by Charnes et al. ( 1978). DEA is a relative, technical efficiency measurement tool, which uses oper ations research techniques to automatically calculate the weights assigned to th e inputs and outputs of the production units being assessed. The actual input/ou tput data values are then multiplied with the calculated weights to determine th e efficiency scores. DEA is a nonparametric multiple criteria method; no product ion, cost, or profit function is estimated from the data. 2.20 Multi-Attribute Fuzzy Integrals When mutual preferential independence among criteria can be assumed, consider th at the utility function is additive and takes the form of a weighted sum. The as sumption of mutual preferential independence among criteria is, however, rarely verified in practice. To be able to take interaction phenomena among criteria in to account, it has been proposed to substitute a monotone set function on attrib utes set N called the fuzzy measure to the weight vector involved in the calcula tion of weighted sums. Such an approach can be regarded as taking into account n ot only the importance of each criterion but also the importance of each subset of criteria. Choquet integral is a natural extension of the weighted arithmetic mean (Grabisch, 1992; Sugeno, 1974).

10 C. Kahraman 3. MULTI-OBJECTIVE DECISION MAKING: A CLASSIFICATION OF METHODS In multiple objective decision making, application functions are established to measure the degree of fulfillment of the decision maker’s requirements (achievemen t of goals, nearness to an ideal point, satisfaction, etc.) on the objective fun ctions and are extensively used in the process of finding “good compromise” solution s. MODM methodologies can be categorized in a variety of ways, such as the form of the model (e.g., linear, nonlinear, or stochastic), characteristic of the dec ision space (e.g., finite or infinite), or solution process (e.g., prior specifi cation of preferences or interactive). Among MODM methods, we can count multi-ob jective linear programming (MOLP) and its variants such as multi-objective stoch astic integer linear programming, interactive MOLP, and mixed 0-1 MOLP; multi-ob jective goal programming (MOGoP); multi-objective geometric programming (MOGeP); multi-objective nonlinear fractional programming; multi-objective dynamic progr amming; and multi-objective genetic programming. The formulations of these progr amming techniques under fuzziness will not be given here since most of them will be explained in detail in the subsequent chapters of this book with numerical e xamples. The intelligent fuzzy multi-criteria decision making methods will be ex plained by Waiel F. Abd El-Wahed in Chapter 2. When a MODM problem is being form ulated, the parameters of objective functions and constraints are normally assig ned by experts. In most real situations, the possible values of these parameters are imprecisely or ambiguously known to the experts. Therefore, it would be mor e appropriate for these parameters to be represented as fuzzy numerical data tha t can be represented by fuzzy numbers. 4. DIFFUSION OF FUZZY SETS INTO MULTICRITERIA DECISION MAKING The classic MADM methods generally assume that all criteria and their respective weights are expressed in crisp values and, thus, that the rating and the rankin g of the alternatives can be carried out without any problem. In a real-world de cision situation, the application of the classic MADM method may face serious pr actical constraints from the criteria perhaps containing imprecision or vaguenes s inherent in the information. In many

MCDM Methods and Fuzzy Sets 11 cases, performance of the criteria can only be expressed qualitatively or by usi ng linguistic terms, which certainly demands a more appropriate method. The most preferable situation for a MADM problem is when all ratings of the criteria and their degree of importance are known precisely, which makes it possible to arra nge them in a crisp ranking. However, many of the decision making problems in th e real world take place in an environment in which the goals, the constraints, a nd the consequences of possible actions are not known precisely (Bellman and Zad eh, 1970). These situations imply that a real decision problem is very complicat ed and thus often seems to be little suited to mathematical modeling because the re is no crisp definition (Zimmermann and Zysno, 1985). Consequently, the ideal condition for a classic MADM problem may not be satisfied, in particular when th e decision situation involves both fuzzy and crisp data. In general, the term “fuz zy” commonly refers to a situation in which the attribute or goal cannot be define d crisply, because of the absence of welldefined boundaries of the set of observ ation to which the description applies. A similar situation is when the availabl e information is not enough to judge or when the crisp value is inadequate to mo del real situations. Unfortunately, the classic MADM methods cannot handle such problems effectively, because they are only suitable for dealing with problems i n which all performances of the criteria are assumed to be known and, thus, can be represented by crisp numbers. The application of the fuzzy set theory in the field of MADM is justified when the intended goals or their attainment cannot be defined or judged crisply but only as fuzzy sets (Zimmermann, 1987). The presen ce of fuzziness or imprecision in a MADM problem will obviously increase the com plexity of the decision situation in many ways. Fuzzy or qualitative data are op erationally more difficult to manipulate than crisp data, and they certainly inc rease the computational requirements in particular during the process of ranking when searching for the preferred alternatives (Chen and Hwang, 1992). Bellman a nd Zadeh (1970) and Zimmermann (1978) introduced fuzzy sets into the MCDM field. They cleared the way for a new family of methods to deal with problems that had been inaccessible to and unsolvable with standard MCDM techniques. Bellman and Zadeh (1970) introduced the first approach regarding decision making in a fuzzy environment. They suggested that fuzzy goals and fuzzy constraints could be defi ned symmetrically as fuzzy sets in the space of alternatives, in which the decis ion was defined as the confluence between the constraints to be met and the goal s to be satisfied. A maximizing decision was then

12 C. Kahraman defined as a point in the space of alternatives at which the membership function of a fuzzy decision attained its maximum value. Baas and Kwakernaak’s (1977) appr oach was widely regarded as the most classic work on the fuzzy MADM method and w as often used as a benchmark for other similar fuzzy decision models. Their appr oach consisted of both phases of MADM, the rating of criteria and the ranking of multiple aspect alternatives using fuzzy sets. Yager (1978) defined the fuzzy s et of a decision as the intersection (conjunction) of all fuzzy goals. The best alternative should possess the highest membership values with respect to all cri teria, but unfortunately, such a situation rarely occurs in the case of a multip le attribute decisionmaking problem. To arrive at the best acceptable alternativ e, he suggested a compromise solution by proposing the combination of max and mi n operators. For the determination of the relative importance of each attribute, he suggested the use of the Saaty method through pairwise comparison based on t he reciprocal matrix. Kickert (1978) summarized the fuzzy set theory application s in MADM problems. Zimmermann’s (1985, 1987) two books include MADM applications. There are a number of very good surveys of fuzzy MCDM (Chen and Hwang, 1992; Fo dor and Roubens, 1994; Luhandjula, 1989; Sakawa, 1993). Dubois and Prade (1980), Zimmermann (1987), Chen and Hwang (1992), and Ribeiro (1996) differentiated the family of fuzzy MADM methods into two main phases. The first phase is generally known as the rating process, dealing with the measurement of performance rating s or the degree of satisfaction with respect to all attributes of each alternati ve. The aggregate rating, indicating the global performance of each alternative, can be obtained through the accomplishment of suitable aggregation operations o f all criteria involved in the decision. The second phase, the ranking of altern atives, is carried out by ordering the existing alternatives according to the re sulted aggregated performance ratings obtained from the first phase. Some titles among recently published papers can show us the latest interest areas of MADM a nd MODM. Ravi and Reddy (1999) rank both coking and noncoking coals of India usi ng fuzzy multi-attribute decision making. They use Saaty’s AHP and Yager’s (1978) fu zzy MADM approach to arrive at the coal field having the best quality coal for i ndustrial use. Fan et al. (2002) propose a new approach to solve the MADM proble m, where the decision maker gives his/her preference on alternatives in a fuzzy relation. To reflect the decision maker’s preference

MCDM Methods and Fuzzy Sets 13 information, an optimization model is constructed to assess the attribute weight s and then to select the most desirable alternatives. Wang and Parkan (2005) inv estigate a MADM problem with fuzzy preference information on alternatives and pr opose an eigenvector method to rank them. Three optimization models are introduc ed to assess the relative importance weights of attributes in a MADM problem, wh ich integrate subjective fuzzy preference relations and objective information in different ways. Omero et al. (2005) deal with the problem of assessing the perf ormance of a set of production units, simultaneously considering different kinds of information, yielded by data envelopment analysis, a qualitative data analys is, and an expert assessment. Hua et al. (2005) develop a fuzzy multiple attribu te decision making (FMADM) method with a three-level hierarchical decision makin g model to evaluate the aggregate risk for green manufacturing projects. Gu and Zhu (2006) construct a fuzzy symmetry matrix by referring to the covariance defi nition of random variables as attribute evaluation space based on a fuzzy decisi on making matrix. They propose a fuzzy AHP method by using the approximate fuzzy eigenvector of such a fuzzy symmetry matrix. This algorithm reflects the disper sed projection of decision information in general. Fan et al. (2004) investigate the multiple attribute decision making (MADM) problems with preference informat ion on alternatives. A new method is proposed to solve the MADM problem, where t he decision maker gives his/her preference on alternatives in a fuzzy relation. To reflect the decision maker’s subjective preference information, a linear goal p rogramming model is constructed to determine the weight vector of attributes and then to rank the alternatives. Ling (2006) presents a fuzzy MADM method in whic h the attribute weights and decision matrix elements (attribute values) are fuzz y variables. Fuzzy arithmetic operations and the expected value operator of fuzz y variables are used to solve the FMADM problem. Xu and Chen (2007) develop an i nteractive method for multiple attribute group decision making in a fuzzy enviro nment. The method can be used in situations where the information about attribut e weights is partly known, the weights of decision makers are expressed in exact numerical values or triangular fuzzy numbers, and the attribute values are tria ngular fuzzy numbers. Chen and Larbani (2006) obtain the weights of a MADM probl em with a fuzzy decision matrix by formulating it as a two-person, zero-sum game with an uncertain payoff matrix. Moreover, the equilibrium solution and the res olution method for the MADM game are developed. These results are validated by a product development example of nano-materials.

14 C. Kahraman Some recently published papers on fuzzy MODM are given as follows: El-Wahed and Abo-Sinna (2001) introduce a solution method based on the theory of fuzzy sets a nd goal programming for MODM problems. The solution method, called hybrid fuzzygoal programming (HFGP), combines and extends the attractive features of both fu zzy set theory and goal programming for MODM problems. The HFGP approach is intr oduced to determine weights to the objectives under the same priorities as using the concept of fuzzy membership functions along with the notion of degree of co nflict among objectives. Also, HFGP converts a MODM problem into a lexicographic goal programming problem by fixing the priorities and aspiration levels appropr iately. Rasmy et al. (2002) introduce an interactive approach for solving MODM p roblems based on linguistic preferences and architecture of a fuzzy expert syste m. They consider the decision maker’s preferences in determining the priorities an d aspiration levels, in addition to analysis of conflict among the goals. The ma in concept is to convert the MODM problem into its equivalent goal programming p roblem by appropriately setting the priority and aspiration level for each objec tive. The conversion approach is based on the fuzzy linguistic preferences of th e decision maker. Borges and Antunes (2002) study the effects of uncertainty on multiple-objective linear programming models by using the concepts of fuzzy set theory. The proposed interactive decision support system is based on the interac tive exploration of the weight space. The comparative analysis of indifference r egions on the various weight spaces (which vary according to intervals of values of the satisfaction degree of objective functions and constraints) enables the study of the stability and evolution of the basis that corresponds to the calcul ated efficient solutions with changes of some model parameters. Luhandjula (1984 ) used a linguistic variable approach to present a procedure for solving the mul tiple objective linear fractional programming problem (MOLFPP). Dutta et al. (19 92) modified the linguistic approach of Luhandjula such as to obtain an efficien t solution to MOLFPP. StancuMinasian and Pop (2003) points out certain shortcomi ngs in the work of Dutta et al. and gives the correct proof of theorem, which va lidates the obtaining of the efficient solutions. We notice that the method pres ented there as a general one does only work efficiently if certain hypotheses (r estrictive enough and hardly verified) are satisfied. Li et al. (2006) improve t he fuzzy compromise approach of Guu and Wu (1999) by automatically computing pro per membership thresholds instead of choosing them. Indeed, in practice, choosin g membership thresholds arbitrarily may result in an infeasible optimization pro blem. Although a minimum satisfaction degree is adjusted to get a fuzzy efficien t

MCDM Methods and Fuzzy Sets 15 solution, it sometimes makes the process of interaction more complicated. To ove rcome this drawback, a theoretically and practically more efficient two-phase ma x–min fuzzy compromise approach is proposed. Wu et al. (2006) develop a new approx imate algorithm for solving fuzzy multiple objective linear programming (FMOLP) problems involving fuzzy parameters in any form of membership functions in both objective functions and constraints. A detailed description and analysis of the algorithm are supplied. Abo-Sinna and Abou-El-Enien (2006) extend the TOPSIS for solving large scale multiple objective programming problems involving fuzzy par ameters. These fuzzy parameters are characterized as fuzzy numbers. For such pro blems, the –Pareto optimality is introduced by extending the ordinary Pareto optim ality on the basis of the –level sets of fuzzy numbers. An interactive fuzzy decis ion-making algorithm for generating an –Pareto optimal solution through the TOPSIS approach is and provided where the decision maker is asked to specify the degre e the relative importance of objectives. 5. CONCLUSIONS The main difference between the MADM and MODM approaches is that MODM concentrat es on continuous decision space aimed at the realization of the best solution, i n which several objective functions are to be achieved simultaneously. The decis ion processes involve searching for the best solution, given a set a conflicting objectives, and thus, a MODM problem is associated with the problem of design f or optimal solutions through mathematical programming. In finding the best feasi ble solution, various interactions within the design constraints that best satis fy the goals must be considered by way of attaining some acceptable levels of se ts of some quantifiable objectives. Conversely, MADM refers to making decisions in the discrete decision spaces and focuses on how to select or to rank differen t predetermined alternatives. Accordingly, a MADM problem can be associated with a problem of choice or ranking of the existing alternatives (Zimmermann, 1985). Having to use crisp values is one of the problematic points in the crisp evalua tion process. As some criteria are difficult to measure by crisp values, they ar e usually neglected during the evaluation. Another reason is about mathematical models that are based on crisp values. These methods cannot deal with decision m akers’ ambiguities, uncertainties, and vagueness that cannot be handled by crisp v alues. The use of fuzzy set

16 C. Kahraman theory allows us to incorporate unquantifiable information, incomplete informati on, non obtainable information, and partially ignorant facts into the decision m odel. When decision data are precisely known, they should not be placed into a f uzzy format in the decision analysis. Applications of fuzzy sets within the fiel d of decision making have, for the most part, consisted of extensions or “fuzzific ations” of the classic theories of decision making. Decisions to be made in comple x contexts, characterized by the presence of multiple evaluation aspects, are no rmally affected by uncertainty, which is essentially from the insufficient and/o r imprecise nature of input data as well as the subjective and evaluative prefer ences of the decision maker. Fuzzy sets have powerful features to be incorporate d into many optimization techniques. Multiple criteria decision making is one of these, and it is certain that more frequently you will see more fuzzy MCDM mode ling and applications in the literature over the next few years. REFERENCES Abo-Sinna, M.A., 2004, Multiple objective (fuzzy) dynamic programming problems: a survey and some applications, Applied Mathematics and Computation, 157: 861–888. Abo-Sinna, M.A., and Abou-El-Enien, T.H.M., 2006, An interactive algorithm for large scale multiple objective programming problems with fuzzy parameters throug h TOPSIS approach, Applied Mathematics and Computation, forthcoming. Baas, S.M., and Kwakernaak, H., 1977, Rating and ranking of multiple-aspect alternatives us ing fuzzy sets, Automatica, 13: 47 58. Bellman, R., and Zadeh, L.A., 1970, Decis ion-making in a fuzzy environment, Management Science, 17B: 141 164. Borges, A.R ., and Antunes, C.H., 2002, A weight space-based approach to fuzzy multipleobjec tive linear programming, Decision Support Systems, 34: 427– 443. Charnes, A., Coop er, W.W., and Rhodes, E., 1978, Measuring the efficiency of decision making unit s, European Journal of Operations Research, 2: 429 444. Chen, S.J., and Hwang, C .L., 1992, Fuzzy Multiple Attribute decision-making, Methods and Applications, L ecture Notes in Economics and Mathematical Systems, 375: Springer, Heidelberg. C hen, Y-W., and Larbani, M., 2006, Two-person zero-sum game approach for fuzzy mu ltiple attribute decision making problems, Fuzzy Sets and Systems, 157: 34–51. Dub ois, D., and Prade, H., 1980, Fuzzy Sets and Systems: Theory and Applications, A cademic Press, New York. Dutta, D., Tiwari, R.N., and Rao, J.R., 1992, Multiple objective linear fractional programming problem—a fuzzy set theoretic approach, Fu zzy Sets and Systems, 52(1): 39–45.

MCDM Methods and Fuzzy Sets 17 El-Wahed, W.F.A., and Abo-Sinna, M.A., 2001, A hybrid fuzzy-goal programming app roach to multiple objective decision making problems, Fuzzy Sets and Systems, 11 9: 71–85. Fan, Z-P., Hu, G-F., and Xiao, S-H., 2004, A method for multiple attribu te decision-making with the fuzzy preference relation on alternatives, Computers and Industrial Engineering, 46: 321–327. Fan, Z-P., Ma, J., and Zhang, Q., 2002, An approach to multiple attribute decision making based on fuzzy preference info rmation on alternatives, Fuzzy Sets and Systems, 131: 101–106. Fodor, J.C., and Ro ubens, M., 1994, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer, Dordrecht. Grabisch, M., 1992, The application of fuzzy integrals in mul ticriteria decision making, European Journal of Operational Research, 89: 445–456. Gu, X., and Zhu, Q., 2006, Fuzzy multi-attribute decision-making method based o n eigenvector of fuzzy attribute evaluation space, Decision Support Systems, 41: 400–410. Guu, S.M., and Wu, Y.K., 1999, Two-phase approach for solving the fuzzy linear programming problems, Fuzzy Sets and Systems, 107: 191–195. Hua, L., Weipin g, C., Zhixin, K., Tungwai, N., and Yuanyuan, L., 2005, Fuzzy multiple attribute decision making for evaluating aggregate risk in green manufacturing, Journal o f Tsinghua Science and Technology, 10(5): 627–632. Hwang, C-L., and Yoon, K., 1981 , Multiple Attribute Decision Making, Lecture Notes in Economics and Mathematica l Systems, Heidelberg, Berlin, Springer-Verlag. Kickert, W.J.M., 1978, Towards a n analysis of linguistic modeling, Fuzzy Sets and Systems, 2(4): 293–308. Li, X., Zhang, B., and Li, H., 2006, Computing efficient solutions to fuzzy multiple obj ective linear programming problems, Fuzzy Sets and Systems, 157: 1328–1332. Ling, Z., 2006, Expected value method for fuzzy multiple attribute decision making, Jo urnal of Tsinghua Science and Technology, 11(1): 102–106. Luhandjula, M.K., 1984, Fuzzy approaches for multiple objective linear fractional optimization, Fuzzy Se ts and Systems, 13(1): 11–23. Luhandjula, M.K., 1989, Fuzzy optimization: an appra isal, Fuzzy Sets and Systems, 30: 257–282. Omero, M., D’Ambrosio, L., Pesenti, R., a nd Ukovich, W., 2005, Multiple-attribute decision support system based on fuzzy logic for performance assessment, European Journal of Operational Research, 160: 710–725. Rasmy, M.H., Lee, S.M., Abd El-Wahed, W.F., Ragab, A.M., and El-Sherbiny , M.M., 2002, An expert system for multiobjective decision making: application o f fuzzy linguistic preferences and goal programming, Fuzzy Sets and Systems, 127 : 209–220. Ravi, V., and Reddy, P.J., 1999, Ranking of Indian coals via fuzzy mult i attribute decision making, Fuzzy Sets and Systems, 103: 369–377. Riberio, R.A., 1996, Fuzzy multiple attribute decision making: a review and new preference elic itation techniques, Fuzzy Sets and Systems, 78: 155–181. Saaty, T.L., 1980, The An alytic Hierarchy Process: Planning, Priority Setting, Resource Allocation, McGra w-Hill, New York. Saaty, T.L., 1996, The Analytic Network Process, RWS Publicati ons, Pittsburgh, PA. Sakawa, M., 1993, Fuzzy Sets and Interactive Multiobjective Optimization, Applied Information Technology, Plenum Press, New York.

18 C. Kahraman Stancu-Minasian, I.M., and Pop, B., 2003, On a fuzzy set approach to solving mul tiple objective linear fractional programming problem, Fuzzy Sets and Systems, 1 34: 397–405. Sugeno, M., 1974, Theory of fuzzy integrals and its applications, PhD thesis, Tokyo Institute of Technology, Tokyo, Japan. Wang, Y-M., and Parkan, C. , 2005, Multiple attribute decision making based on fuzzy preference information on alternatives: ranking and weighting, Fuzzy Sets and Systems, 153: 331–346. Wu, F., Lu, J., and Zhang, G., 2006, A new approximate algorithm for solving multip le objective linear programming problems with fuzzy parameters, Applied Mathemat ics and Computation, 174: 524–544. Xu, Z-S., and Chen, J., 2007, An interactive me thod for fuzzy multiple attribute group decision making, Information Sciences, 1 77(1): 248–263. Yager, R.R., 1978, Fuzzy decision making including unequal objecti ves, Fuzzy Sets and Systems, 1: 87–95. Zadeh, L.A., 1965, Fuzzy sets, Information and Control, 8: 338–353. Zimmermann, H.J., 1978, Fuzzy programming and linear prog ramming with several objective functions, Fuzzy Sets and Systems, 1: 45–55. Zimmer mann, H.J., 1985, Fuzzy Set Theory and Its Applications, Kluwer, Nijhoff Publish ing, Boston. Zimmermann, H.J., 1987, Fuzzy Sets, Decision Making, and Expert Sys tems, Kluwer, Boston. Zimmermann, H.J., and Zysno, P., 1985, Quantifying vaguene ss in decision model, European Journal of Operational Research, 22: 148–158.

INTELLIGENT FUZZY MULTI-CRITERIA DECISION MAKING: REVIEW AND ANALYSIS Waiel F. Abd El-Wahed Operations Researchs and Decisison Support Department, Faculty of Computers & In formation, Menoufia University, Shiben El-Kom, Egypt Abstract: This chapter highlights the implementation of artificial intelligence techniques to solve different problems of fuzzy multi-criteria decision making. The reason s behind this implementation are clarified. In additions, the role of each techn ique in handling such problem are studied and analyzed. Then, some of the future research work is marked up as a guide for researchers who are working in this r esearch area. Intelligent optimization, fuzzy multi-criteria decision making, re search directions Key words: 1. 1.1 INTRODUCTION Mathematical Model of Fuzzy Multi-Criteria Decision Making Multi-criteria decision making (MCDM) represents an interest area of research si nce most real-life problems have a set of conflict objectives. MCDM has its root s in late-nineteenth-century welfare economics, in the works of Edgeworth and Pa reto. A mathematical model of the MCDM can be written as follows: Min s Z [z1 (x), z2 (x),..., z K (x)]T (1) C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 19

20 W.F. Abd El-Wahed where S {x X Ax b, x Rn , x 0} where: Z(x) = C x is the K-dimensional vector of objective functions and C is th e vector of cost corresponding to each objective function, S is the feasible reg ion that is bounded by the given set of constraints, A is the matrix of technica l coefficients of the left-hand side of constraints, b is the right-hand side of constraints (i.e., the available resources), x is the n-dimensional vector of d ecision variables. When the objective functions and constraints are linear, then the model is a linear multi-objective optimization problem (LMOOP). But, if any objective function and/or constraints are nonlinear, then the problem is descri bed as a nonlinear multi-objective optimization problem (NLMOOP). Since problem (1) is deterministic, it can be solved by using different approaches such as fol lows: 1. 2. 3. 4. Utility function approach, Interactive programming, Goal progr amming, and Fuzzy programming. But, in the real world, the input information to model (1) may be vague, for exa mple, the technical coefficient matrix (A) and/or the available resource values (b) and/or the coefficients of objective functions (C). Also, in other situation s, the vagueness may exist, such as the aspiration levels of goals (zi(x)) and t he preference information during the interactive process. All of these cases lea d to a fuzzy multi-criteria model that can be written as follows: Min Z S [z1 (x), z2 (x),..., z K (x)]T (2) where ~ ~ ~ S {x X A x b, x Rn , x 0} .

Intelligent Fuzzy MCDM: Review and Analysis 21 This fuzzy model is transformed into crisp (deterministic) by implementing an ap propriate membership function. So, the model can be classified into two classes. If any of the objective functions, constraints, and membership functions are li near, then the model will be LFMOOP. But, if any of the objective functions and/ or constraints and/or membership functions are nonlinear, then the model is desc ribed as NLFMOOP. Different approaches can handle the solution of problem (2). A ll of these approaches depend on transforming problem (2) from fuzzy model to cr isp model via determining an appropriate membership function that is the backbon e of fuzzy programming. Definition 1.1: Fuzzy set Let X denote a universal set. Then a fuzzy subset à of X is defined by its membership function: A: X [0,1] (3) That assigns to each element x X a real number in the interval [0, 1] and Ã(x) rep resents the grade of membership function of x in A. The main strategy for solvin g model (2) can be handled according to the following scheme: Step 1. Examine th e type of preference information needed. Step 2. If a priori articulation of pre ference information is available use, one of the following programming schemes: 2.1 Fuzzy goal programming, 2.2 Fuzzy global criterion, or 2.3 Another appropria te fuzzy programming technique. Otherwise, go to step (3). Step 3. If progressiv e articulation of preference information is available, use the following program ming scheme: 3.1 Fuzzy interactive programming, 3.2 Interactive fuzzy goal progr amming, or 3.3 Another appropriate fuzzy interactive programming technique. Step 4. End strategy. Each programming scheme involved different solution methodolog ies that will be indicated in Section 1.3.

22 W.F. Abd El-Wahed 1.2 Historical Background of Fuzzy MCDM In 1970, Bellman and Zadah highlighted the main pillar of fuzzy decision making that can be summarized as follows: D G C (4) where G is the fuzzy goal, C is the fuzzy constraints, and D is the fuzzy decisi on that is characterized by a suitable membership function as follows: D ( x) min( G ( x), C ( x)) . (5) The maximizing decision is then defined as follows: max x X D ( x) max min ( x X G ( x ), C ( x )) . (6) For k fuzzy goals and m fuzzy constraints, the fuzzy decision is defined as foll

ows: D G1 G2 ... Gk C1 C2 ... Cm (7) and the corresponding maximizing decision is written as follows: max x X D ( x) max min ( x X G1 ( x ),..., Gk , C1 ( x )..., Cm ( x )) . (8) For more details about this point, see Sakawa (1993). Since this date, many rese arch works have been developed. In this section, the light will be focused on a sample of research works on FMCDM from the last 25 years to extract the main sho rtcomings that argue for us to direct attention toward the intelligent technique s as an alternative methodology for overcoming these drawbacks. In FMCDM problem s, the membership function depends on where the fuzziness existed. If the fuzzin ess in the objective functions coefficients, the membership function may be repr esented by

Intelligent Fuzzy MCDM: Review and Analysis 23 1 k k ( Z ( x )) if k Z k ( x) Lk Lk , Uk U k Z (x ) U k Lk if Z k ( x) (9) Uk 0 if Z k ( x ) where Uk is the worst upper bound and Lk is the best lower bou nd of the objective function k, respectively. They are calculated as follows: Uk Lk (Z k ) max (Z k ) min max Z k (x ) x X min Z k (x ), k x X (10) 1, 2,. . ., K If the fuzziness is existed in the right-hand side of the constraints, the const raints are transformed into equalities and then the following membership functio n is applied (Lai and Hwang, 1996): [(Ax)i k (Z k (x)) [(bi 0 (bi di )] / di if if (bi bi di ) ( Ax)i di ) ( Ax)i (b i bi , di ) (bi di ) di ) (Ax)i ] / di (11) if (bi ( Ax)i or ( Ax)i where the membership function is assumed to be symmetrically triangular function s. The problem solver may assume any other membership function based on his/her experience. Besides, some mathematical and statistical methods develop a specifi c membership function. On the other side, the intelligent techniques provide the problem solver with a powerful techniques to create or estimate these functions as will be indicated later. If we assumed that the FMCDM problem has fuzzy obje ctive functions, then the deterministic model of the FMCDM is written as follows : max subject to µk (Z k (x)) , k 1,2,..., K n (12)

aij x j j 1 bi , i 1,2,..., m 1,2,...,m; j 1,2,..., n, k 1,2,..., K xj 0 0, i 1

24 W.F. Abd El-Wahed where is an auxiliary variable and can be worked at a satisfaction level. Model (7) can be solved as a single objective linear/nonlinear programming problem. Af ter the Bellman and Zadah paper, several research studies were adopted, such as Hannan (1983) and Zimmerman (1987) who handled fuzzy linear programming with mul tiple objectives by assuming a special form of the membership function. Hannan a ssumed discrete membership function, and Zimmerman used a continuous membership function. Boender (1989), Sakawa (1993), and Baptistella and Ollero (1980) imple mented the fuzzy set theory in interactive multi-criteria decision making. For m ore historical information, see Sakawa (1993) and Lai and Hwang (1996). Also, se e Biswal (1992), Bhattacharya et al. (1992), Bit (1992), Boender et al. (1989), Buckley (1987), Lothar and Markstrom (1990) for more solution methodologies. Man y real-life problems have been formulated as FMCDM and have been solved by using an appropriate technique. Some of these applications involved production, manuf acturing, location allocation problems, environmental management, business, mark eting, agriculture economics, machine control, engineering applications and regr ession modeling. A good classification with details can be found in Lai and Hwan g (1996). A new literature review (Zopounidis and Doumpos, 2002) assures the sam e field of applications. 1.3 Shortcomings of the FMCDM Solution Approaches The problems that meet either the solution space construction or the model devel opment can be classified into three categories as follows: 1) illstructured, 2) semi-well structured or, 3) well structured. Each category has been characterize d by specific criteria to indicate its class. Some of these indicator criteria o f ill-structured problems are as follows: 1. 2. 3. 4. 5. 6. There is no availabl e solution technique to solve the model. There is no standard mathematical model to represent the problem. There is no ability to involve the qualitative factor s in the model. There is no available solution space to pick up the optimal solu tion. There is a difficulty in measuring the quality of the result solution(s). There is kind of vagueness of the available information that leads to complexity in considering it into the model account.

Intelligent Fuzzy MCDM: Review and Analysis 25 If some of these criteria exist, then the problem will belong to the second cate gory, which is called semi-ill-structured problems. But, if all of these criteri a and others do not exist, then the problem will belong to the third category, w hich is called well-structured problems. It is clear that there is no problem re garding the third category. Fortunately, the first and second categories represe nt a rich area for investigation, especially in the era of information technolog y where all the sciences are interchanged in a complex manner to a degree that o ne can find difficulty in separating between sciences. In other words, biologica l sciences, sociology, insects’ science, and so on attracted researchers to simula te them by using computer technology that consequently reflects its positive pro gress on the optimization research work. Let us now apply these criteria of illstructured problems on FMCDM problems. For FMCDM model structure, the following problems are represented as an optical stone to more progress in this area. Some of these problems are as follows: 1. Incorporating fuzzy preferences in the mod el still needs new methodologies to take the model into account without increasi ng the model complexity. 2. Right now, the FMCDM models are transformed into cri sp models to solve it by using the available traditional techniques. This transf ormation reduces both the efficiency and the effectiveness of the fuzzy solution methodologies. So, we need to look for a new representation methodology to incr ease or at least keep the efficiency of the fuzzy methodology. 3. As mentioned a bove, the membership function is the cornerstone of fuzzy programming, and right now, the problem solvers assumed it according the experience. As a result, the solution will be different according to the selected membership function. This w ill lead to another problem, which is which solution is better or qualifies more for the problem under study. In this case, there is an invitation to implement the progress in information technology to discover an appropriate membership fun ction. 4. Large-scale FMCDM models still need more research especially when inco rporating large preference information. Regarding the solution methodologies, th ere are some difficulties in enhancing them. Some of them are: 1. Some of the ex isting ranking approaches that have been used to solve the FMCDM problem are not perfect.

26 W.F. Abd El-Wahed 2. Fuzzy integer programming with multi-criteria can be considered a combinatori al optimization problem, and as a result, it needs an exponential time algorithm to go with it. 3. In 0-1 FMCDM problems (whatever small scale or large scale), the testing process of the Pareto-optimal solution is considered the NP-hard pro blem. 4. In FMCDM problems, a class of problems exist that are known as the glob al convex problems, where the good solutions in the objective space are similar to those in the decision space. So, we need a new methodology to perform well wi th them. 5. In fuzzy and nonfuzzy MCDM problems, there is a difficulty in constr ucting an initial solution that should be close to the Paretooptimal solution to reduce the solution time. So, we need powerful methodology based information te chnology to deal with this problem. Because of these shortcomings and others, FM CDM attracts the attentions of researchers to enhance the field of FMCDM by deve loping more powerful links (bridges) between it and other sciences. In this chap ter, we will highlight the link between artificial intelligence and FMCDM to ove rcome all or some of the mentioned problems. This link leads to a new and intere sting area of research called “intelligent optimization.” The general strategy for t he integration between artificial intelligence (AI) techniques and FMCDM problem s may be done according the following flowchart seen in Figure 2. In the next su bsection, some of the intelligent techniques will be introduced briefly. 1.4 Some Intelligent Techniques AI is the branch of computer technology that simulates the human behavior via in telligent machines to perform well and better than humans. Computer science rese archers are wondering how to extract their ideas from the biological systems of human beings such as thinking strategies, the nervous system, and genetics. AI a lso extends to the kingdom of insects such as the ant colony. The tree that summ arizes the different commercial forms of AI techniques is shown in Figure 1. Eac h AI technique can perform well in specific situations more so than in others. F or example, expert systems (ESs) can handle the qualitative factors or preferenc es that can not be included in the mathematical model. Artificial neural network s (ANNs) are successfully applied in prediction, classification, pattern and voi ce recognition, and so on. Simulated annealing

Intelligent Fuzzy MCDM: Review and Analysis 27 (SA), genetic algorithms (GA), and particle swarm optimization (PSO) are used as stochastic search methods to deal with multi-criteria combinatorial optimizatio n problems. The implementation of AI techniques to handle different problems in FMCDM depends on the following conditions: 1. The nature of the problem that FMC DM suffers from, 2. The availability of the solution techniques and its performa nce, 3. The environmental factors that affect the problem under study. AI techni ques can be classified according to their functions as follows: 1. Symbolic proc essing, where the knowledge is treated symbolically not numerically. In other wo rds, the process is not algorithmical. These techniques are ES, fuzzy expert sys tem (FES), and decision support system (DSS). 2. Search methods that are impleme nted to search and scan the large solution space of combinatorial optimization p roblem. These techniques are able to pick an acceptable or preferred solution in less time compared with the traditional solution procedures. Examples of these search methods are GA, SA, ant colony optimization (ACO), PSO, DNA computing, an d any hybrid of them. 3. Learning process that is responsible for doing forecast ing, classifications, and function estimating based on enough historical data ab out the problem under study. These techniques are ANN and neurofuzzy systems. No w, we shall classify the intelligent FMCDM problems based upon the implemented t echnique. 1.4.1 Expert System and FMCDM ES is an intelligent computer program that consists of three modules: 1) inferen ce engine module, 2) knowledge-base module, and 3) userinterface module. This sy stem can produce one of the following functions: 1) conclusion, 2) recommendatio n, and 3) advice. The main feature of the ES is its ability to treat the problem s symbolically not algorithmically. So, it can perform a good job regarding both the decision maker’s preferences and the qualitative factors that cannot be inclu ded in the mathematical model because of its increase in the degree of model com plexity.

28 W.F. Abd El-Wahed DSS Expert Systems (ESs) Artificial Neural Network (ANN) Parallel hybrid expert systems Simulated Annealing (SA) Parallel Intelligent Systems Artificial Intelligence Techniques (AIT) Genetic Algorithms (GAs) Hybrid Intelligent Systems Ant Colony Optimization ACO Scatter Search DNA Computing Particle Swarm Optimization (PSO) Figure 1. The tree diagram of artificial intelligence techniques Generally speaking, ES has been applied to solve different applications that can be modeled in MCDM. For example, Lothar and Markstrom (1990) developed an exper t system for a regional planning system to optimize the industrial structure of an area. In this system, AI paradigms and numeric multi-criteria optimization te chniques are combined to arrive at a hybrid approach to discrete alternative sel ection. These techniques include 1) qualitative analysis, 2) various statistical checks and recommendations, 3) robustness and sensitivity analysis, and 4) help for defining acceptable regions for analysis. Jones et al. (1998) developed an intelligent system called “GPSYS” to deal with linear and integer goal programming. The intelligent goal programming system is one that is designed to allow a nonsp ecialist access to, and clear understanding of a goal programming solution and a nalysis techniques. GPSYS has an analysis tool such as Pareto

Intelligent Fuzzy MCDM: Review and Analysis Defuzzify the developed model of FMCDM problem. 29 Classes of AI Techniques Symbolic Processing Select or develop the suitable membership function. Learning Process Consider the DM’s preferences Solve the model by using an appreciate technique. Search Methods No Evaluate the solution. Yes End Solution Figure 2. The integration between AI techniques and FMCDM phases detection and restoration, normalization, automated lexicographic redundancy che cking, and an interactive facility. Abd El-Wahed (1993) developed a decision sup port system with a goal programming based ES to solve engineering problems. In t his research, the statistical analysis and the decision maker’s preferences are co mbined in an ES to assign the differential weights of the sub-goals in goal prog ramming problems. Also, Rasmy et al. (2001) presented a fuzzy ES to include the qualitative factors that could not be involved in the mathematical model of the multi-criteria assignment problem in the field of bank processing. The approach depends on evaluating the model solution by using the developed fuzzy ES. If the solution is coincided with the evaluation criteria, the approach is terminated. Otherwise, some modification on the preferences is done in the feedback to reso lve the model again and so on until getting a solution coincides with the evalua tion criteria. Little research work regarding FMCDM has been done. For example, Rasmy et al. (2002) presented an interactive approach

30 W.F. Abd El-Wahed for solving the MCDM problem with fuzzy preferences in both aspiration level det ermination and priority structure by using the framework of the fuzzy expert sys tem. The main idea of this approach is to convert the MCDM problem into its equi valent goal programming model by setting the aspiration levels and priority of e ach objective function based on fuzzy linguistic variables. This conversion make s the implementation of ES easy and effective. Liu and Chen (1995) present an in tegrated machine troubleshooting expert system (IMTES) that enhances the efficie ncy of the diagnostic process. The role of fuzzy multi-attribute decision-making in ES is determined to be the most efficient diagnostic process, and it creates a “meta knowledge base” to control the diagnosis process. The results of an update search in some available database sites regarding the combination of both ES and FMCDM can be summarized as follows: 1. The mutual integration between ES and MC DM/FMCDM is a rich area for more research, 2. The implementation of ES for deali ng with the problems of FMCDM still needs more research, 3. The combination of E S and other AI techniques needs more research to gain the advantages of both of them in solving the problems of FMCDM problems. The researchers are invited to i nvestigate the following points where they are not covered right now: 1. Applyin g ES to guide the determination process of the aspiration levels of fuzzy goal p rogramming. 2. Applying ES to handle the DM’s preferences in solving interactive F MCDM to reduce the solution time and the solution efforts. 3. Implementing the E S in ranking approaches that have been used to solve FMCDM problems to include t he environmental qualitative factors. 4. Handling ES in solving large-scale FMCD M problems. 5. Combining ES with both parametric analysis and sensitivity analys is to pick a more practical solution.

Intelligent Fuzzy MCDM: Review and Analysis 31 1.4.2 ANN and FMCDM Problems ANN is a simulation of a human nervous system. The ANN simulator depends on the Third Law of Newton: “For any action there is an equal reaction with negative dire ction.” A new branch of computer science is opened for research called “neural compu ting.” Neural computing has been viewed as a promising tool to solve problems that involve large date/preferences or what is called in optimization large-scale op timization problems. Also, the transformation of FMCDM into crisp model needs an appropriate membership function. In other situations, ANN is implemented to sol ve the FMCDM problems without the need to defuzzify the mathematical model of FM CDM problems. ANN offers an excellent methodology for estimating continuous or d iscrete membership functions/ values. To do that, an enormous amount of historic al data is needed to train and test the ANN as well as to get the right paramete rs and topology of it to solve such a problem. On the other side, the complex co mbinatorial FMCDM problems (NP hard problems) may be not represented in a standa rd mathematical form. As a result, ANN can be used to simulate the problem for t he purpose of getting an approximate solution based on a simulator. The main pro blem facing those who are working in this area is the development of the energy (activation) function, which is the central process unit of any ANN. This functi on should have the inherited characteristics of both the objective function and the constraints to train and test the network. There are many standard forms of it such as the sigmoid function and the hyperbolic function. The problem solver must elect a suitable one from them such that can be fitted with the nature of t he problem under study. For the FMCDM with fuzzy objective functions [model (7)] , the energy function can be established by using the Lagrange multiplier method as follows: n E ( x, , , ) t ( µk ( Z k (x)) ) t ( j 1 aij x j bi ) (11) and are the Lagrange multipliers. is the vector of slack where variables. By tak ing the partial derivative of an equation with respect to x, , and , we obtain t he following differential equations:

32 W.F. Abd El-Wahed E/ x E/ E/ x E (x , , , ) E (x , , , ) E (x , , , ) (12) where is called a learning parameter. By setting the penalty parameters and , th e adaptive learning parameters , and initial solution xj(0), then we can solve t he system (9) to obtain . Previous research works use ANN to solve some optimiza tion problems as well as FMCDM specifically. These works can be classified accor ding to the type of treating method of the FMCDM model as follows: 1.4.2.1 Treat ing the Fuzzy Preferences in MCDM Problems For example, Wang (1993) presented a feed-forward ANN approach with a dynamic training procedure to solve multi-crite ria cutting parameter optimization in the presence of fuzzy preferences. In this approach, the decision maker’s preferences are modeled by using fuzzy preference information based on ANN. Wang and Archer (1994) modeled the uncertainty of mult i-criteria, multi-persons decision making by using fuzzy characteristics. They i mplemented the back propagation learning algorithm under monotonic function cons traints. Stam et al. (1996) presented two approaches of ANNs to process the pref erence ratings, which resulted from analytical, hierarchy process, pair-wise com parison matrices. The first approach, implements ANN to determine the eigenvecto rs of the pair-wise comparison matrices. This approach is not capable of general izing the preference information. So, it is not appropriate for approximating th e preference ratings if the decision maker’s judgments are imprecise. The second a pproach uses the feed-forward ANN to approximate accurately the preference ratin gs. The results show that this approach is working well with respect to imprecis e pair-wise judgments. Chen and Lin (2003) developed the decision neural network (DNN) to use in capturing and representing the decision maker’s preferences. Then , with DNN, an optimization problem is solved to look for the most desirable sol ution. 1.4.2.2 Handling Fuzziness in FMCDM Models It is clear that ANN is capabl e of solving the constrained optimization problems, especially the applications that require on-line optimization. Gen et al. (1998) discussed a two-phase appro ach to solve MCDM problems with fuzziness in both objectives and constraints. Th e main proposed steps to solve the FMCDM model (2) can be summarized as follows:

Intelligent Fuzzy MCDM: Review and Analysis 33 1. Construct the membership function based on positive ideal and negative ideal (worst values) solutions. 2. Apply the concept of -level cut, where [0,1] to tra nsform the model into a crisp model. 3. Develop the crisp linear programming mod el based on steps (1) and (2). 4. According to the augmented Lagrange multiplier method, we can create the Lagrangian function to transform the result model in step (3) into an unconstrained optimization problem. The Lagrangian function is implemented as an energy (activation) function to activate the developed ANN. 5. If the DM accepts the solution, stop. Otherwise, change and go to the step (1). The results show that the result solution is close to the best compromise solut ion that has been calculated from the two-phase approach. The method has an adva ntage; if the decision maker is not satisfied with the obtained solutions, he/sh e can get the best solutions by changing the -level cut. 1.4.2.3 Determining the Membership Functions Ostermark (1999) proposed a fuzzy ANN to generate the memb ership functions to new data. The learning process is reflected in the shape of the membership functions, which allows the dynamic adjustment of the functions d uring the training process. The adopted fuzzy ANN is applied successfully to mul ti-group classification-based multi-criteria analysis in the economical field. 1 .4.2.4 Searching the Solution Space of Ill-Structured FMCDM Problems Gholamian e t al. (2005) studied the application of hybrid intelligent system based on both fuzzy rule and ANN to: Guide the decision maker toward the noninferior solutions . Support the decision maker in the selection phase after finishing the search p rocess to analyze different noninferior points and to select the best ones based on the desired goal levels. The idea behind developing this system is the ill-s tructured real-world problem in marketing problems where the objective can not b e expressed in a mathematical form but in the form of a set of historical data. This

34 W.F. Abd El-Wahed means that ANN can do well with respect to any other approach. From the above an alysis, we can deduce that many research points are still uncovered. It means th at the integration area between ANN and FMCDM is very rich for more research. Th ese points are summarized as follows: 1. Applying the ANN to solve FMCDM problem s in its fuzzy environment without transforming it into a crisp model to obtain more accurate, efficient, and realistic solution(s). 2. Developing more approach es to enhance the process of generating real membership functions. 3. Studying t he effect of using different membership functions on the solution quality and pe rformance. 4. Implementing the ANN to solve more large-scale FMCDM problems that represented the real-life case. 5. Combining both ES and ANN to develop more po werful approaches to consider the preference information (whatever quantitative/ qualitative) in FMCDM problems. 6. Applying the ANN to do both parametric and se nsitivity analysis of the real-life problems that can be represented by the FMCD M model. 1.4.3 Tabu Search A tabu search (TS) was initiated by Glover as an iterative intelligent search te chnique capable of overcoming the local optimality when solving the CO problems. The search process is based on a neighborhood mechanism. The neighborhood of a solution is defined as a set of all formations that can be obtained by a move th at is a process for transforming the search from the current solution to its nei ghboring solution. If the move is not listed on the TS, the move is called an “adm issible move.” If the produced solution at any move is better than all enumerated solutions in prior iterations, then this solution is saved as the best one. The candidate solutions, at each iteration, are checked by using the following tabu conditions: 1. Frequency memory that is responsible for keeping the knowledge of how the same solutions have been determined in the past. 2. Recency memory that prevents cycles of length less than or equal to a predetermined number of itera tions. TS has an important property that enables it to avoid removing the powerf ul solutions from consideration. This property depends on an element called an a spiration mechanism. This element means that if the TS

Intelligent Fuzzy MCDM: Review and Analysis 35 list captured a solution with a value strictly better than the best obtained so far, the TS can stop. TS is applied to solve some FMCDM problems. For example, B agis (2003) proposed a new approach based on TS to determine the membership func tions of a fuzzy logic controller. The simulation results indicated that the giv en approach is performed well, and as a result it is effective in determining su ch a membership function. Li et al. (2004) presented a TS method as a stochastic global optimization method for solving very large combinatorial optimization ta sks and for extending a continuous-valued function for the fuzzy optimization pr oblems. They approved the performance of the proposed method by applying it to a n elementary fuzzy optimization problem such as the method for fuzzy linear prog ramming; fuzzy regression and the training of fuzzy neural networks are also pre sented. Choobineh et al. (2006) proposed an algorithm to deal with a sequencing of n-jobs on a single machine with sequence-dependent setup times and m-objectiv e functions. The algorithm generates a set of solutions that reflects the object ives’ weights and close to the best observed values of the objectives. In addition , the authors formulated a mixed integer linear program to obtain the optimal so lution of a tripleobjective functions problem. Most of the published research wo rks have not focused on FMCDM problems. 1.4.4 Simulated Annealing (SA) The SA algorithm is a search technique designed to look for a global minimum amo ng many local minima. The algorithm simulates the thermodynamic process of annea ling metals by slow cooling where at high temperatures, molecules in metal move rapidly with respect to each other. If the metal is slow cooled sufficiently, th en thermal mobility is lost. The resulting arrangement of atoms tends to form a pure crystal that is completely ordered. This ordered state occurs when the syst em has achieved minimum energy by an annealing process that must be cooled suffi ciently slowly to reach thermal equilibrium. The SA search method is a powerful tool to provide excellent solutions of single objective optimization problems to reduce the computational cost. Later, this approach was adapted for the multi-o bjective framework by Serafini (1985), Czy ak et al. (1994) and Ulungu et al. (1 995). But they examined only the notion of the probability in the multi-objectiv e framework. Serafini (1985) used simulated annealing on the multiobjective fram ework. Czy ak and Jaszkiewicz (1998) and Ulungu et al. (1998) designed a complet e MOSA algorithm and tested it with a multi-

36 W.F. Abd El-Wahed objective combinatorial optimization problem. Ulungu et al. (1999) presented an interactive version of MOSA to solve an industrial application problem. Suppapit narm et al. (2000) proposed a different simulated annealing approach to handle m ulti-objective problems. Czy ak et al. (1994) hybridized both SA and GA to provi de efficient solutions of multi-objective optimization problems. Loukil et al. ( 2006) proposed a multi-objective SA algorithm to tackle a production scheduling problem in a flexible job-shop with particular constraints such as batch product ion; production of several sub-products followed by assembly of the final produc t, and possible overlaps for the processing periods of two successive operations of the same job. For more details in this area of research, see both Suman (200 2) and (2003). In the literature, there are some research works regarding MCDM p roblems, and the available fuzzy research works are under the general title “fuzzy optimization” not specific FMCDM problems. So, this area of research is ripe for more investigations. 1.4.5 Genetic Algorithms and FMCDM The GA is a search algorithm that mimics the processes of natural evolution. The problem addressed by GA is searching the solution space is to identify the best problems that are combinatorial or large scale or illstructured in general. GA encodes the variables of problems in either binary or real-valued vectors. Each code is called a chromosome. In binary coding there are two decoding functions t o convert from real to binary and vice versa. In addition, mutation, crossover, and selection are the three important operators used for generating a new soluti on within the solution space. For example, the mutation operator introduces new genetic material into the population. Crossover recombines individuals to create new individuals. The selection process elects the next generation by using 1) t ournament selection, 2) proportional selection, 3) ranking selection, 4) steadystate selection, and 5) manual selection. An evaluation function called the “fitne ss function” is generated to test the result solution. In the case of constrained optimization problems, Lagrange multipliers are used to transform the problem in to an unconstrained optimization problem to be used as a fitness function. The g eneral flowchart of a GA for solving an optimization problem is shown in Figure 3.

Intelligent Fuzzy MCDM: Review and Analysis Select the coding method. 37 Binary - code 11111100011 Represent the solutions via the selected coding method. Real - code 1234567891 I nitialize the GA by creating randomly a solution that satisfies all constraints of the FMCDM problem. Evaluate the solution via the developed fitness function. Apply GA operators to generate new solution(s). Is the DM satisfied? No Interact with DM to achieve his / her desires. Yes Stop Figure 3. General schema of GA to solve FMCDM problems GAs seem desirable for solving MOOPs because they deal simultaneously with a set of solutions (the so-called population) that allows the problem solver to find several members of the Pareto optimal set in a single run of the algorithm, inst ead of having to perform a series of separate runs, such as with the traditional mathematical programming techniques. Additionally, GAs are less susceptible to the shape or continuity of the Pareto front, whereas these two issues are a real concern for mathematical programming techniques. The integration between GA and MOOPs can be classified in the following two categories: Non-Pareto Techniques Under this category, we will consider approaches that do not incorporate directl y the concept of Pareto optimality. Although these approaches are efficient, mos t of them enable us to produce certain portions of the Pareto front. However, th eir simplicity has made them

38 W.F. Abd El-Wahed popular among a certain sector of researchers. These approaches are as follows: 1. 2. 3. 4. Aggregating approaches, Lexicographic ordering, The -constraint meth od, and Target-vector approaches. Pareto-Based Techniques In this category, the main idea is finding the set of st rings in the population that are Pareto nondominated by the rest of the populati on. These strings are assigned the highest rank and are eliminated from addition al considerations. Another set of Pareto nondominated strings are determined fro m the remaining population and are assigned the next highest rank. Some of the a pproaches that implement this idea are: 1. 2. 3. 4. Pure Pareto ranking, Multi-o bjective genetic algorithm (MOGA), Nondominated sorting genetic algorithm (NSGA) , and Nondominated pareto genetic algorithm (NPGA). In the context of this chapter, some works have been found and can be classified into the following categories: 1.4.5.1 Interactive FMCDM-Based GA Sakawa and ot hers presented a series of papers in this category. The ideas of these works can be summarized in the following: Kato et al. (1997) introduce an interactive sat isfying method using GA for getting the satisfying solution for a decision maker from an extended Pareto optimal solution set. In this method, for a certain val ue of -level cut and reference membership function, the solution of large-scale multi-objective 0-1 programming is obtained by adopting a GA with decomposition procedures. Sakawa and Yauchi (1999) highlight the multi-objective, nonconvex, n onlinear programming problems with fuzzy goals and solve it by applying an inter active fuzzy satisfying method. In this method, the Pareto optimal solution is o btained by solving the augmented mini-max problem for which the floating point G A called GENOCOP III is applicable.

Intelligent Fuzzy MCDM: Review and Analysis 39 Sakawa and Yauchi (2000) proposed an interactive decision-making method for solv ing multi-objective, nonconvex programming problems with fuzzy numbers through c o-evolutionary GAs. In this paper, the authors were trying to overcome the drawb acks of GENCOP III by introducing a method to generate an initial feasible point and a bisection method. This modification leads to a new GENCOP called revised GENCOP III. Sakawa and Kubota (2000) solved an application in job shop schedulin g with fuzzy processing time and fuzzy due date by using GA. Sakawa and Kato (20 02) deal with the general multi-objective 0-1 programming problems that involve positive and negative coefficients. The extended GA with double strings is imple mented with a new decoding algorithm for individuals. The double strings map eac h individual to a feasible solution based on backtracking and individual modific ation. For more details about the GA and FMCDM, see Sakawa (2002). Basu (2004) a pplied an interactive fuzzy satisfying method based on an evolutionary programmi ng technique for short-term multi-objective hydrothermal scheduling. The multi-o bjective problem is formulated by assuming that the decision maker has fuzzy goa ls for each of the objective functions and that the evolutionary programming tec hniquebased fuzzy satisfying method is applied for generating a corresponding op timal noninferior solution for the decision maker’s goals. Wahed et al. (2005) pre sented a contribution in this area by suggesting an interactive approach to dete rmine the preferred compromise solution for the MCDM problems in the presence of fuzzy preferences. Here, the decision maker evaluates the solution by using a d efined set of linguistic variables, and consequently, the achievement membership function can be constructed for each objective function. The used nonnegative d ifferential weights are determined based on the entropy degree of each objective function to support transforming the MCDM into a single objective function. 1.4 .5.2 Goal Programming-Based GAs Goal programming (GP) is an important technique that is capable of solving a problem with multiple goals. The concept of goal pr ogramming (GP) is extended to solve multi-objective decision-making problems bec ause of its ability to transform it into a single-objective programming problem with or without priority through putting the objective functions as goal constra ints with predetermined aspiration levels. Also, FGP is extended to solve the co mplex problems in MCDM/FMCDM problems,

40 W.F. Abd El-Wahed especially with implementing GAs. In this case, some research works have been en umerated as follows: Zheng et al. (1996) discussed the initialization process, f itness function structure, and the GA operators in the proposed GA for solving n onlinear goal programming (NLGP). Gen et al. (1997) developed a GA to solve fuzz y NLGP. They assumed that the implemented membership functions are strictly mono tone decreasing (or increasing) and continuous functions with the set of objecti ve functions and certain maximum tolerance limits to the given resources. Hu et al. (2007) suggested a method for generating the solution that is consistent wit h the decision maker’s desires where the goal with high priority may have the firs t level of goal achievement. The method uses a co-evolutionary genetic algorithm to solve the nonlinear, nonconvex problem that results from the original proble m. GENCOPIII package is used to handle this problem. 1.4.5.3 Fuzzy Programming-B ased GAs Li et al. (1997) presented an improved GA for solving a multi-objective solid transportation problem with consideration of the coefficients of the obje ctive function as fuzzy numbers. The selection and evaluation process in GA are done by incorporating ranking of fuzzy numbers with integral value. Kim (1998) d esigned a two-phase genetic algorithm to improve the system performance in nonli near and complex problems. The first phase is responsible for generating a fuzzy rule base that covers as many of the training examples as possible. The second phase constructed fine-tuned membership functions that minimize the system error . Liu and Iwamura (2001) provide a fuzzy simulation-based GA to handle both fuzz y objectives and goal constraints as well as other ideas. Jimenez et al. (2003) proposed an evolutionary algorithm to solve fuzzy nonlinear programming as a fir st step to solving the general nonlinear programming problem. Sasaki and Gen (20 03) proposed a GA for solving fuzzy multiple objective design problems by implem enting a new chromosomes representation that makes the GA more effective. Wang e t al. (2005) implemented the multi-objective GA to extract interpretable fuzzy r ule-based knowledge from data where the genes

Intelligent Fuzzy MCDM: Review and Analysis 41 are arranged into control genes and parameter genes. This division enables the f uzzy sets and rules to be optimally reduced. At the end of this section, we can decide that the implementation of GAs in solving the FMCDM problems are occupied a wide interest of the research move so than any other AI searches technique. F or more knowledge, see the following website: http://www.jeo.org/emo/ EMOOjourna ls.html. However, there are still some problems in FMCDM problems that have not been studied yet such as: 1. Large-scale FMCDM problems with fuzzy numbers in th e objective functions and constraints. 2. Combining both ES and GA to handle the fuzzy preferences in MCDM problems to get a more powerful solution method. 3. I mplementing the GA to study both sensitivity and parametric analysis of linear a nd nonlinear FMCDM. 1.4.6 Ant Colony Optimization Ant colony optimization (ACO) is a meta-heuristic approach that emulates the for aging behavior of real ants to find the shortest paths between food sources and their nest. This approach is proposed by Dorigo (1992). During the ant’s walk from food sources and vice versa, ants deposit a chemical substance called “Pheromone” o n the ground to guide the rest of ants to the shortest and safest path they shou ld follow. The artificial ants that simulate the real ants perform random walks on a completely connected graph G = (S, L), whose vertices are the solution comp onents S and the connections L. This graph is based on probabilistic model calle d the “Pheromone model.” When a constrained combinatorial optimization problem is co nsidered, the constraints are built into the ants to get the feasible solution(s ) only. ACO methods have been successfully applied to diverse combinatorial opti mization problems, including traveling salesman, quadratic assignment, vehicle r outing, telecommunication networks, graph coloring, constraint satisfaction, Ham iltonian graphs, and scheduling (Cordon et al., 2002). The following chart indic ated the mechanism of ACO in solving combinatorial optimization (CO). The ACO ap proach is performing well in combinatorial network optimization problems where t he solution space is difficult to enumerate especially in large-scale problems. It has been applied to solve the multiobjective combinatorial optimization probl ems. For example, Chan and Swarnkar (2006) present a fuzzy goal programming appr oach to model the

42 W.F. Abd El-Wahed ACO Solution components CO problem Pheromone model Probabilistic solution constructi on Pheromone value update Initialization of pheromone Figure 4. Mechanism of ACO in solving combinatorial optimization (Blum, 2005) machine tool selection and operation allocation problem of flexible manufacturin g systems. The proposed model is optimized by an ant colony algorithm to the com putational complexities involved in solving the problem. Doerner et al. (2006) a pplied Pareto ant colony optimization (P-ACO) that performs particularly well fo r integer linear programming. The given procedure identifies several efficient p ortfolio solutions within a few seconds and correspondingly initializes the pher omone trails before running P-ACO. This extension offers a larger exploration of the search space at the beginning of the search with low cost. Marc Gravel et a l. (2002) applied the ACO for getting the solution of an industrial scheduling p roblem in an aluminum casting center. They present an efficient representation s cheme of a continuous horizontal casting process that takes into account several objectives that are important to the scheduler. A little research work has been done in using ACO and MCDM/ FMCDM problems. Most of the research work is done i n multi-objective combinatorial optimization problems (MOCOPs) since the meta-he uristics perform much better than the other approaches. So, this area needs more and more research especially in combinatorial FMCDM problems. 1.4.7 Particle Sw arm Optimization (PSO) The basic principles of PSO are represented by a set of moving particles that is initially thrown inside the search space. Each particle is characterized by the following features: 1. A position and a velocity, 2. It knows its position and the objective function value for this position,

Intelligent Fuzzy MCDM: Review and Analysis 43 3. It knows its neighbors, the best previous position, and the objective functio n value, 4. It remembers its best previous position, 5. It is considered that th e neighborhood of a particle includes this particle itself. At each time step, t he behavior of a given particle is a compromise between three possible choices: 1. Following its own way, 2. Going toward its best previous position, 3. Going t oward the best neighbor’s best previous position. The basic equations of PSO can b e formalized as follows: vt xt with 1 1 c1vt xt c 2 p i ,t vt 1 xt c 3 p g ,t xt (13) vt : xt : Pit : Pgt : velocity at time step t , position at time step t , best previous position at ti me step t , best neighbours previous best, at time step t , (or best neighbor), c1 , c2 , c3 : social/cognitive confidence coefficients. PSO has been used in solving some real-life applications that involved multi-obj ectives. For example, Parsopoulos and Vrahatis (2002) presented the first study on MCDM by using PSO algorithm. The authors highlighted some important issues su ch as: 1. The ability of PSO to obtain the Pareto optimal points as well as the shape of the Pareto front. 2. Applying the weighted sum approach with fixed or a daptive weights. 3. Adopting the well-known GA approach VEGA for MCDM problems t o the PSO framework to develop multi-swarm PSO to be implemented in MCDM problem s in an effective manner.

44 W.F. Abd El-Wahed The study can be considered the corner stone of applying PSO to solve such MCDM problems. Salman et al. (2002) proposed a PSO to task assignment. The PSO system combines local search methods (through selfexperience) with global search metho ds (through neighboring experience), attempting to balance exploration and explo itation. A scan of some international electronic databases indicated that PSO ha s not applied yet in solving FMCDM problems. 1.5 Conclusions From the above analysis, one can conclude that the implementation of AI techniqu es to handle FMCDM problems has occupied a reasonable amount of attention from t he researchers with respect to some AI techniques such as ES, ANN, and GAs. But other techniques have not been opened yet such as SA, TS, PSO, DNA, and parallel hybrid techniques for handling the problems of FMCDM. However, the AI technique s that have been applied proved that they have the following advantages when dea ling with FMCDM problems: 1. They have the possibility to consider the qualitati ve factors in the model structure and the solution procedure. 2. They can handle the decision maker’s preferences, which are characterized as fuzzy preferences. 3 . They can deal with a large amount of data that can be used in solving FMCDM pr oblems. 4. The availability to estimate the aspiration levels in FMCDM. 5. The a bility to estimate (determine) the membership functions that can be implemented to transform the FMCDM problem into a crisp problem to be handled easily. 6. The possibility to search and scan the search space in fuzzy multicriteria combinat orial optimization problems where the search space is very large. 7. The AI tech niques successes in solving different real-life problems such as scheduling, man ufacturing, chemical, managerial, and other industrial applications. 1.5.1 Resea rch Directions The future research direction in this area is viewed from two angles:

Intelligent Fuzzy MCDM: Review and Analysis 45 1. Improving the performance of intelligent techniques by combining two or more of these techniques to get more powerful ones. 2. Implementing the available tec hniques to handle the FMCDM problems. We shall talk about each individual case. First: Improving the available techniques: a) The mathematical background of the se techniques needs more investigation and analysis. b) Extending the AI techniq ues to handle more problems regarding FMCDM. c) Studying the possibility and val idity of combining more than two of these techniques to outperform the original ones. d) Developing a comparative study between the AI techniques (metaheuristic techniques) to measure the performance of each one with respect to others. On t he other side, measuring the performance and/or the quality of the solution(s) w hen changing the parameters of each technique. e) Lights should be placed on new hybrid techniques as well as on parallel hybrid techniques that will be probabl y perform better than the AI techniques themselves. Second: Intelligent FMCDM re search directions: This area of research still needs intensive research such as the following directions: a) Large-scale FMCDM with mixed integer decision varia bles needs more investigation especially by using parallel hybrid intelligent sy stems to reduce the solution time. b) Measuring the performance of AI techniques in higher dimensional FMCDM problems where the only test of performance is usin g benchmark functions. In addition, the theoretical analysis of measuring AI per formance needs a look from the researchers. c) Developing the theoretical analys is to deal with the FMCDM problems in its fuzzy environment without transforming it into crisp model, where the resulting solution may be more reasonable than t he solution results from the transformation process. d) Studying the effect of c hanging the AI techniques parameters on the solution behavior of FMCDM problems. In other words, understanding

46 W.F. Abd El-Wahed the dynamics of swarm’s dynamics (as in PSO) and the Pheromones dynamics (as in AC O) on the behavior of the optimization process. e) Until now, no one has tried t o open the area on doing both parametric and sensitivity analysis of MCDM and/or FMCDM by applying the AI techniques. The time is suitable for performing intell igent parametric analysis of MCDM and/or FMCDM problems. The results may be bett er than the traditional techniques for both linear and nonlinear FMCDM problems. As an idea, conduct the study of intelligent parametric analysis based on satis fying Kuhn Tucker conditions or look for another easy way to do that. f) Develop ing an intelligent system that combined most AI techniques to deal with FMCDM pr oblems. For example, ES, ANN, SA, GA, and PSO may be combined in the following m anner: ES may handle the fuzzy preferences and other qualitative factors that ha ve a great impact on the FMCDM problem behavior. This phase can be used as an ev aluation process of the result solution(s). Applying GA as a second phase to sca n the solution space to get a satisfactory Pareto optimal solution. Improving th e performance of a PSO-based ANN with SA to use the GA output as an initial solu tion to this phase as a trial to obtain a better solution than the one in step ( b). This is a proposed scenario, and the researchers can change this scenario in different manners. More attention can be paid to measure the performance, and e ffectiveness should be done to compare the results with the existing techniques. 1. The ANN (for example) can be used to generate a reasonable membership functi on for solving the FMCDM problems based on the desires of the DM and/or the hist orical data of the problem. 2. Applying the AI techniques to implement the ranki ng approaches to deal with FMCDM problems. 3. Developing new approaches based on AI techniques to handle the fuzzy multi-attribute decision-making problems wher e a little research work has been done in this area. 4. Implementing AI techniqu es to solve FMCDM in the presence of multiple decision makers with indifference preferences information. 5. Invoking AI techniques in both interactive and goal programming to solve FMCDM. For example, developing an ANN to capture and repres ent the decision maker’s preferences to support the search process for obtaining t he most desirable solution.

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FUZZY ANALYTIC HIERARCHY PROCESS AND ITS APPLICATION Tufan Demirel1, Nihan Çetin Demirel1, and Cengiz Kahraman2 Yildiz Technical University, Department of Industrial Engineering, Yildiz-Istanb ul Turkey Istanbul Technical University, Department of Industrial Engineering, B esiktas-Istanbul Turkey 2 1 Abstract: The analytic hierarchy process (AHP) is one of the most widely-used multiattribu te decision-making methods. In this section we overview the fuzzy AHP methods ex isting in the literature. We present the four different approaches of fuzzy AHP methods by giving numerical examples. Multi-attribute decision-making, fuzzy AHP , extent analysis, entropy value Key words: 1. INTRODUCTION The analytic hierarchy process (AHP) is one of the most widely-used multi-attrib ute decision-making (MADM) methods. In any planning and decision-making process, a systematic and logical approach is used to arrive at the solution. In the mul ti-criteria decision analysis, the fuzzy set theory might be the most common met hod in dealing with uncertainty. The analytic hierarchy process has been used in many different fields as a multi-attribute decision analysis tool with multiple alternatives and criteria. AHP uses “pair-wise comparisons” and matrix algebra to w eight criteria. The decision is made by using the derived weights of the evaluat ive criteria (Saaty, 1980). Importance is measured on an integer-valued 1 9 scal e, with each number having the interpretation shown in Table 1. C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 53

54 T. Demirel et al. In this chapter, we give the literature review results in the following section. Section 2.1 presents an introduction and a definition of fuzzy AHP. Sections 2. 2, 2.4, 2.6, and 2.9 present Van Laarhoven and Pedrycz’s approach, Buckley’s fuzzy A HP, Chang’s extent analysis method, and fuzzy AHP with entropy value, with numeric al examples, respectively. The last section summarizes suggestions for additonal research. Table 1. Interpretation of Entities in a Pair-wise Comparison Matrix Value of ai j Interpretation 1 3 5 7 9 2, 4, 6, 8 Objectives i and j have equal importance Objective i is weakly more important th an objective j Experience and judgment indicate that objective i is strongly mor e important than objective j Objective i is very strongly or demonstrably more i mportant than objective j Objective i is absolutely more important than objectiv e j Intermediate values 2. LITERATURE REVIEW Many fuzzy AHP methods are proposed by various authors. These methods are system atic approaches to the alternative selection and justification problem by using the concepts of fuzzy set theory (Zadeh, 1965) and hierarchical structure analys is. Decision makers usually find that it is more confident to give interval judg ments than fixed value judgments. Because usually he/she cannot be explicit abou t his/her preferences because of the fuzzy nature of the comparison process. The earliest work in fuzzy AHP appeared in van Laarhoven and Pedrycz (1983), which compared fuzzy ratios described by triangular membership functions. Buckley (198 5) determines fuzzy priorities of comparison ratios membership functions trapezo idal. Stam et al. (1996) explore how recently developed artificial intelligence techniques can be used to determine or approximate the preference ratings in AHP . They conclude that the feed-forward neural network formulation appears to be a powerful tool for analyzing discrete alternative multi-criteria decision proble ms with imprecise or fuzzy ratio-scale preference judgments. Chang (1996) introd uces a new approach for handling fuzzy AHP, with the use of triangular fuzzy num bers for pair-wise comparison scale off fuzzy AHP and the use of the extent anal ysis method for the synthetic extent values of

Fuzzy AHP and Its Application 55 the pair-wise comparisons. Cheng (1997) proposes a new algorithm for evaluating naval tactical missile systems by the fuzzy analytical hierarchy process based o n grade value of membership function. Weck et al. (1997) present a method to eva luate different production cycle alternatives adding the mathematics off fuzzy l ogic to the classic AHP. Any production cycle evaluated in this manner yields a fuzzy set. The outcome of the analysis can finally be defuzzified by forming the surface center of gravity of any fuzzy set, and the alternative production cycl es investigated can be ranked in terms of the main objective set. Kahraman et al . (1998) use a fuzzy objective and subjective method obtaining the weights from AHP and make a fuzzy weighted evaluation. Cheng et al. (1999) propose a new meth od for evaluating weapon systems by analytical hierarchy process based on lingui stic variable weight. Zhu et al. (1999) make a discussion on extent analysis met hod and applications of fuzzy AHP. Badri (2001) proposed a combined AHP-GP model for quality control systems. Creed (2001), Jansen et al. (2001) and Martinez-To me et al. (2000) investigate food industry, customer satisfaction and food suppl y chain. Cebeci (2001) and Cebeci and Kahraman (2002) proposed a fuzzy AHP model to Measure customer satisfaction of catering service companies. Yu (2002) incor porates an absolute term linearization technique and a fuzzy rating expression i nto a GP-AHP model for solving group decision-making fuzzy AHP problem. Kahraman et al. (2004) provide an analytical tool to select the best Turkish catering fi rm providing the most customer satisfaction. The fuzzy analytic hierarchy proces s is used to compare three Turkish catering firms in their paper. Tolga et al. ( 2005) aim at creating an operating system selection framework for decision maker s. Since decision makers have to consider both economic and noneconomic aspects of technology selection, both factors are considered in their developed framewor k. They develop the economic part of the decision process by fuzzy replacement a nalysis. Noneconomic factors and financial figures are combined using a fuzzy an alytic hierarchy process approach. Hsiao and Chou (2006) propose a gestalt-like perceptual measure method by combining gestalt grouping principles and fuzzy ent ropy. The purpose of the proposed method is not to evaluate the grades of altern atives but to measure the gestalt-like perceptual degrees for home page design. They identify the weights using fuzzy AHP.

56 T. Demirel et al. 2.1 Fuzzy AHP Inability of AHP to deal with the imprecision and subjectiveness in the pair-wis e comparison process has been improved in fuzzy AHP. Instead of a crisp value, f uzzy AHP uses a range of value to incorporate the decision maker’s uncertainty (Ku swandari, 2004). 2.2 Van Laarhoven and Pedrycz’s Approach (1983) Van Laarhoven and Pedrycz (1983) offer an algorithm that is the direct extension of Saaty’s AHP method. They identify the weights through the AHP operations. In t hat study, Laarhoven and Pedrycz use the triangular fuzzy numbers. The computati on steps are the same as those in crisp AHP. The Lootsma’s logarithmic least-squar es method is used to derive fuzzy weights and fuzzy performance scores (Chen et al., 1992). Laarhoven and Pedrycz’s approach is shown by the following steps: Step 1. Consult with the MCDMs and obtain n+1 fuzzy reciprocal matrix that takes the following form as shown (1). ~ a 121 ~ a 122 ~ a 12 P 12 1 ,1 ,1 ~ a 211 ~ a 212 ~ a 21 P 21 ~ a1n1 ~ a1n 2 ~ a 1 nP 1n ~ A 1 ,1 ,1 ~ a2n1 ~ a2n2 ~ a 2 nP 2n (1) ~ a n 11 ~ a n 12 ~ an1P n1 ~ a n 21 ~ a n 22 ~ an2 P n2 1 ,1 ,1 ~ where aijP ij are fuzzy ratios estimated by multiple decision makers. Note tha t pij may be 0 when no decision maker expresses his/her comparison ratios or gre ater than 1 when more than one decision maker expresses his/her comparison ratio s.

Fuzzy AHP and Its Application 57 Step 2. Let z i n l i , mi , u i . Solve the following linear equations: n n Pij li j 1 j i Pij j 1 j 1 Pij u j j 1 k 1 j 1 ln lijk , i (2) n n n Pij mi j 1 j i Pij j 1 j i Pij m j j 1 j i ln mijk , i (3) n n n Pij ui j 1 j i Pij

j 1 j i Pij l j j 1 j i ln uijk , i (4) As ln lijk and ln uijk are lower and upper values of ln aijk the following must hold true [see Eq. (2)]: ln a jik , ln l ijk ln l jik ln u ijk ln u jik 0, i, j, k. (5) Thus Eqs. (2) and (4) are linear dependent. The same holds for Eq. (3). Generall y, a solution for Eqs. (2), (3), and (4) is given as: zi li t1 , mi t2 , ui t1 , i (6) where t1 and t2 can be chosen arbitrarily. Step 3. The right sides of the equati ons above are operated using logarithmic operations. Then we obtain the fuzzy we ight in Eq. (7): wi where n 1 i 1 1 exp li , 2

exp mi , 3exp ui (7) 1 n 2 i 1 1 n 3 i 1 1 exp ui exp mi exp li

58 T. Demirel et al. Equation (7) can also be used to determine the performance score rij . Step 4. S teps 1 3 are repeated several times until all reciprocal matrices are solved. Wi th the fuzzy weights and performance scores, we can calculate the fuzzy utility for alternative Ai as n ui j 1 w j rij (8) 2.3 A Numerical Example A company is looking for a sales manager. There are four applicants for this pos ition. The company is also looking for four attributes from these applicants. Th ese attributes are leadership, mathematic creativity, communication skill, and e xperimentation. Figure 1 shows the hierarchy of sales manager selection problem. Three decision makers will be graded for the four attributes. Figure 1. The hierarchy of the sales manager selection The three decision makers’ opinions about the relative importance of a pair of att ributes are shown in Tables 2 to 6.

Fuzzy AHP and Its Application Table 2. Pair-Wise Comparisons of Applicants for Leadership A1 A1 (1, 1, 1) (3/2, 2, 5/2) (3/2, 2, 5/2) (2/3, 1, 3/2) (7/2, 4, 9/2) (7/2, 4, 9/2) (5/2, 3, 7/2) A2 (2/5, 1/2, 2/3) (2/5, 1/2, 2/3) A3 (2/3, 1, 3/2) (5/2, 3, 7/2) (3/2, 2, 5/2) (1, 1, 1) (3/2, 2, 5/2) (2/3, 1, 3/2) A4 (2/9, 1/4, 2/7) (2/ 9, 1/4, 2/7) (2/7, 1/3, 2/5) (2/5, 1/2, 2/3) (2/3, 1, 3/2) (2/3, 1, 3/2) (2/5, 1 /2, 2/3) (2/3, 1, 3/2) (1, 1, 1) 59 A2 A3 A4 (1, 1, 1) (2/7, 1/3, 2/5) (2/5, 1/2, 2/3) (3/2, 2, 5/2) (2/3, 1, 3/2) (2/3, 1, 3 /2) Table 3. Pair-Wise Comparisons of Applicants for Mathematic Creativity A1 A1 A2 A3 (1, 1, 1) (2/5, 1/2, 2/3) (2/5, 1/2, 2/3) (5/2, 3, 7/2) (7/2, 4, 9/2 ) (7/2, 4, 9/2) (2/3, 1, 3/2) A2 (3/2, 2, 5/2) (3/2, 2, 5/2) (1, 1, 1) (3/2, 2, 5/2) (3/2, 2, 5/2) (2/3, 1, 3/2) A3 (2/7, 1/3, 2/5) (2/9, 1/4, 2/7) (2/9, 1/4, 2 /7) (2/5, 1/2, 2/3) (1, 1, 1) (2/7, 1/3, 2/5) (2/7, 1/3, 2/5) (2/5, 1/2, 2/3) A4 (2/3, 1, 3/2) (2/5, 1/2, 2/3) (2/3, 1, 3/2) (5/2, 3, 7/2) (5/2, 3, 7/2) (3/2, 2 , 5/2) (1, 1, 1) A4 Table 4. Pair-Wise Comparisons of Applicants for Communication Skill A1 A1 A2 A3 A4 (1, 1, 1) (2/7, 1/3, 2/5) (2/7, 1/3, 2/5) (2/7, 1/3, 2/5) (2/3, 1 , 3/2) A2 (5/2, 3, 7/2) (5/2, 3, 7/2) (1, 1, 1) (2/3, 1, 3/2) (2/5, 1/2, 2/3) (2 /5, 1/2, 2/3) A3 (5/2, 3, 7/2) (2/3, 1, 3/2) (1, 1, 1) (5/2, 3, 7/2) A4 (2/3, 1, 3/2) (3/2, 2, 5/2) (3/2, 2, 5/2) (2/7, 1/3, 2/5) (1, 1, 1)

60 T. Demirel et al. Table 5. Pair-Wise Comparisons of Applicants for Experimentation A1 A1 A2 (1, 1, 1) (2/3, 1, 3/2) (2/9, 1/4, 2/7) (2/7, 1/3, 2/5) (3/2, 2, 5/2) A 2 (2/3, 1, 3/2) (1, 1, 1) (2/5, 1/2, 2/3) (2/7, 1/3, 2/5) (2/7, 1/3, 2/5) (2/3, 1, 3/2) A3 (7/2, 4, 9/2) (5/2, 3, 7/2) (3/2, 2, 5/2) (5/2, 3, 7/2) (5/2, 3, 7/2) (1, 1, 1) (5/2, 3, 7/2) A4 (2/5, 1/2, 2/3) (2/3, 1, 3/2) A3 A4 (2/7, 1/3, 2/5) (1, 1, 1) Table 6. Pair-Wise Comparisons of Attributes X1 X1 (1, 1, 1) (2/5, 1/2, 2/3) (2/5, 1/2, 2/3) (2/3, 1, 3/2) (2/9, 1/4, 2/7) (2 /3, 1, 3/2) X2 (3/2, 2, 5/2) (3/2, 2, 5/2) (2/3, 1, 3/2) (1, 1, 1) (2/5, 1/2, 2/ 3) (3/2, 2, 5/2) (2/3, 1, 3/2) X3 (7/2, 4, 9/2) X4 (2/3, 1, 3/2) (2/5, 1/2, 2/3) (2/3, 1, 3/2) (2/9, 1/4, 2/7) (1, 1, 1) X2 X3 X4 (3/2, 2, 5/2) (1, 1, 1) (7/2, 4, 9/2) The following phases are taken to solve the problem: 4 4 4 Pij l1 j 2 P1 j j 2 P1 j u j j 2 k 1 P 12 ln l1 jk P 13 P 14 l1 P 12 P 13 P 14 P u2 12 P u3 13 P u4 14 k 1 ln l12 k k 1 ln l13k k 1 ln l14 k

where P12 = 2 (two decision makers) P13 = 1 (one decision maker) P14 = 3 (three decision makers)

Fuzzy AHP and Its Application 61 6l1 2u2 4 u3 3u4 4 ln 2 / 5 ln 2 / 5 ln 2 / 3 ln 2 / 9 ln 2 / 9 ln 2 / 7 4 Pij l2 j 1 j 2 P2 j j 1 j 2 P2 j u j j 1 k 1 j 2 ln l 2 jk l 2 P21 P21 k 1 P23 P24 P23 k 1 P21u1 P24 k 1 P23u 3 ln l 24 k P24 u 4 ln l 21k ln l 23k where P21 = 2 (two decision makers) P23 = 2 (two decision makers) P24 = 3 (three decision makers) 7l 2 2u1 2u3 3u 4 ln(3/2) ln(3/2) ln(5/2) ln(3/2) ln(2/5) ln(2/3) ln(2/3)

By a similar process, obtained linear equations can be represented as 6l1 – 2u2 – 1u 3 – 3u4 = 6.4989 7l2 – 2u1 – 2u3 – 3u4 = 0.4054 5l3 – 1u1 – 2u2 – 2u4 = 3.8962 8l4 – 3u1 – = 3.0163 6m1 – 2m2 – 1m3 – 3m4 = 5.2574 7m2 – 2m1 – 2m3 – 3m4 = 0.4054 5m3 – 1m1 – 2m2 – 2 8962 8m4 – 3m1 – 3m2 – 2m3 = 3.0163 6u1 – 2l2 – 1l3 – 3l4 = 3.8272 7u2 – 2l1 – 2l3 – 3l4 = 5u3 – 1l1 – 2l2 – 2l4 = 0.9162 8u4 – 3l1 – 3l2 – 2l3 = 7.3098

62 T. Demirel et al. The solutions to these equations are given in Table 7: Table 7. The Solutions to the Equations I 1 2 3 4 li 0 0.7870 0.1936 0.9751 mi 0 0.8919 0.2849 1.0629 ui 0.1443 1.1028 0 .5216 1.2572 The exponentials of li, mi, and ui are given in Table 8: Table 8. The Exponentials of li, mi, and ui I 1 2 3 4 exp(li) 1.0000 2.1967 1.2136 2.6514 exp(mi) 1.0000 2.4397 1.3296 2.894 7 exp(ui) 1.1552 3.0125 1.6847 3.5155 We can calculate the fuzzy performance score r11 using Eq. (7) with the exponent ial numbers. r11 = ( 1exp(l1), The terms 1, 2, 4 1 i 1 2exp(m1), 3exp(u1)) and exp u i 3 are computed as 1 4 0.1067 , 2 i 1 exp mi 1 0.1304 4 3 i 1 exp l i 1 0.1416 . The fuzzy performance scores r1j , j:1,2,3,4, can be summarized as r11 = (0.1076 , 0.1304, 0.1635) r12 = (0.2343, 0.3181, 0.4265) r13 = (0.1294, 0.1733, 0.2385) r14 = (0.2829, 0.3774, 0.4977).

Fuzzy AHP and Its Application 63 Steps 1 through 3 are applied to Tables 3, 4, 5, and 6. All results are given in Table 9. Table 9. All Results X1 A1 A2 A3 A4 (0.1067, 0.1304, 0.1635) (0.2343, 0.3181, 0.4265) (0.1295, 0.1733 , 0.2385) (0.2829, 0.3774, 0.4977) X2 (0.1434, 0.1793, 0.2225) (0.1035, 0.1288, 0.1680) (0.4457, 0.5049, 0.5568) (0.1413, 0.1868, 0.2495) X3 (0.3498, 0.4363, 0. 5362) (0.1708, 0.2190, 0.2787) (0.1042, 0.1313, 0.1685) (0.1641, 0.2130, 0.2827) X4 (0.2018, 0.2729, 0.3730) (0.1868, 0.2760, 0.4037) (0.0855, 0.0974, 0.1139) ( 0.2556, 0.3530, 0.4772) W = [(0.2579, 0.3509, 0.4703), (0.1609, 0.2199, 0.3054), (0.0812, 0.0932, 0.1101 ), (0.2418, 0.3354, 0.4612)] We can calculate fuzzy utilities U1, U2, U3, and U4 by Eq. (8) as: U1 = (0.1277, 0.2173, 0.3759) U2 = (0.1361, 0.2529, 0.4687) U3 = (0.1342, 0.2167, 0.3532) U4 = (0.1708, 0.3117, 0.5614). The fuzzy utilities can be ranked by any appropriate fuzzy ranking method. 2.4 Buckley’s (1985) Fuzzy AHP Buckley also extended Saaty’s AHP method to incorporate fuzzy comparison ratios ai j . He pointed out that Van Laarhoven and Pedrycz’s (1983) method was subject to t wo problems. First, the linear equations of obtained equations do not always hav e a unique solution. Second, they insist on obtaining triangular fuzzy numbers f or their weights. Buckley’s (1985) approach is shown in the following steps. Step 1. Consult the decision maker, and obtain the comparison matrix A whose elements are ~ ( aij , bij , cij , d ij ) , where all i and j are trapezoidal tij fuzzy numbers. Step 2. The fuzzy weights wi can be calculated as follows. The geometri c mean for each row is determined as

64 n j 1 T. Demirel et al. 1/ n ~ zi ~ tij , for all i (9) The fuzzy weight wi is given as wi ~ zi n j 1 1 ~ zj . (10) In the following discussion, we will detail the derivation of fuzzy weight wi . Let the left leg and right leg of ~ be, respectively, defined as tij n 1/ n fi j 1 bij aij aij 1/ n , 0,1 (11) n gi j 1 cij d ij

bij , 0,1 . (12) Furthermore, let n 1/ n ai j 1 ~ tij (13) and m a i 1 ai . (14) Similarly, we can define bi and b, ci and c, and d i and d. The fuzzy weight wi is determined as wi ai bi ci d i , , , , a b c d i (15)

Fuzzy AHP and Its Application 65 where the membership function wi x is defined as follows: Let x be a real number on the horizontal axis. The wi x can be summarized as in Table 10. Table 10. Interpretation of Entities in a Pair-wise Comparison Matrix X wi x ai d ai d bi ci , c b a i bi , d c ci d i , b a a i bi , or x d c 0 0 1 0 ,1 0 ,1 When x ci d i , , the x is calculated as b a x m fi ( ) g ( ) , g i ( ) f ( ), m if x if x ai d , bi c ci b, d i a (16) where f ( ) i 1 f i ( ) and g ( ) i 1 gi ( ) ’ Step 2 is repeated for all the fuzzy performance scores. Step 3. The fuzzy weigh ts and fuzzy performance scores are aggregated. The fuzzy utilities Ui, i, are o btained based on n Ui j 1 w j rij , i. (17)

66 T. Demirel et al. 2.5 A Numerical Example A ceramic factory is looking for a general manager. There are three applicants f or this position. The company is also looking for four attributes from these app licants. These attributes are leadership, problem-solving skill, communication s kill, and experimentation. An expert will be graded for the four attributes. The expert opinions about the relative importance of a pair of attributes are shown in Tables 11 to 15. Table 11. Pair-Wise Comparison of Applicants for Leadership A1 (1, 1, 1, 1) (1/3, 1/2, 1/2, 1) (1/4, 1/4, 1/2, 1/2) A2 (1, 2, 2, 3) (1, 1, 1 , 1) (1/3, 1/2, 1/2, 1) A3 (2, 2, 4, 4) (1, 2, 2, 3) (1, 1, 1, 1) A1 A2 A3 Table 12. Pair-Wise Comparison of Applicants for Leadership A1 (1, 1, 1, 1) (2, 3, 3, 4) (1/2, 1/2, 1, 1) A2 (1/4, 1/3, 1/3, 1/2) (1, 1, 1, 1) (1/4, 1/4, 1/3, 1/3) A3 (1, 1, 2, 2) (3, 3, 4, 4) (1, 1, 1, 1) A1 A2 A3 Table 13. Pair-Wise Comparison of Applicants for Leadership A1 (1, 1, 1, 1) (1/7, 1/7, 1/6, 1/6) (1/4, 1/4, 1/3, 1/3) A2 (6, 6, 7, 7) (1, 1, 1, 1) (1, 1, 2, 2) A3 (3, 3, 4, 4) (1/2, 1/2, 1, 1) (1, 1, 1, 1) A1 A2 A3 Table 14. Pair-Wise Comparison of Applicants for Leadership A1 (1, 1, 1, 1) (5, 6, 6, 7) (1/2, 1/2, 1, 1) A2 (1/7, 1/6, 1/6, 1/5) (1, 1, 1, 1) (1/4, 1/4, 1/3, 1/3) A3 (1, 1, 2, 2) (1, 2, 2, 3) (1, 1, 1, 1) A1 A2 A3 Table 15. Pair-Wise Comparison of Attributes X1 X2 X3 X4 X1 X2 X3 X4 (1, 1, 1, 1) (1/3, 1/2, 1/2, 1) (1/3, 1/3, 1/2, 1/2) (3, 3, 3, 3) (1, 2, 2, 3) (1, 1, 1, 1) (1/2, 1/2, 1, 1) (2, 3, 3, 4) (2, 2, 3, 3) (1, 1, 2, 2) (1, 1, 1, 1) (2, 2, 2, 2) (1/3, 1/3, 1/3, 1/3) (1/2, 1/3, 1/3, 1/2) (1/2, 1/2, 1/2, 1/2) (1, 1, 1, 1)

Fuzzy AHP and Its Application 67 The following phases are taken to solve the problem: For the first reciprocal ma trix, the geometric mean is 3 1/ 3 a1 j 1 a1 j 1/ 3 a11 a 12 a 13 1 1 2 1/ 3 1.2599 3 a2 j 1 a2 j 1/ 3 a21 a 22 a 23 1/ 3 1 1 1/ 3 0.6933 3 a3 j 1 a3 j 3 a31 a 32 a 33 1/ 4 1/ 3 1 1/ 3 0.4367 Hence, a i 1 ai 1.2599 0.6933 0.4367 2.3899 .

Similarly, we can get bi and b, ci and c, and di and d. They are summarized as i n Table 16. Table 16. Geometric Means I ai bi ci di 1 1.2599 1.5874 2 2.2894 2 0.6933 1 1 1.4422 3 0.4367 0.5 0.6299 0 .7937 Sum of the k th row ai = 2.3899 bi = 3.0874 ci = 3.6299 di = 4.5253 Thus, (a, b, c, d) = (2.3899, 3.0874, 3.6299, 4.5253). The performance scores rj 1 , j 1, 2, and 3 can be obtained as r11 a1 b1 c1 d1 , , , d c b a a2 b2 c2 d 2 , , , d c b a 0.2784, 0.4373, 0.6477, 0.9579 r21 0.1532, 0.2754, 0.3238, 0.6034

68 T. Demirel et al. r31 a3 b3 c3 d 3 , , , d c b a 0.0965, 0.1377, 0.2040, 0.3321 The performance scores rj 2 , r j 3 , and wi can be obtained as r12 = (0.1495, 0 .1797, 0.2668, 0.3333) r22 = (0.4312, 0.5393, 0.6994, 0.8550) r32 = (0.1186, 0.1 296, 0.2118, 0.2352) r13 = (0.5875, 0.5875, 0.8283, 0.8283) r23 = (0.0930, 0.093 0, 0.1501, 0.1501) r33 = (0.1412, 0.1412, 0.2383, 0.2383) r14 = (0.1170, 0.1288, 0.1888, 0.2111) r24 = (0.5521, 0.6136, 0.7857, 0.8703) r34 = (0.1119, 0.1170, 0 .1888, 0.1987) w1 = (0.1725, 0.2278, 0.2758, 0.3427) w2 = (0.1025, 0.1354, 0.176 2, 0.2604) w3 = (0.1025, 0.1139, 0.1640, 0.1841) w4 = (0.3554, 0.4367, 0.4778, 0 .5765) w1 * r11 a1 , a2 L1 , L2 , b1b2 , c1c2 , d1d 2 R1 , R2 w1 = (a2, b2, c2, d2) L2 = a2(b1 a1) + a1(b2 a2) R2 = [d2(d1 c1) + d1(d2 c2)] Table 17. The Values of wjr1j r11 = (a1, b1, c1, d1) , L1 = (b1 a1)(b2 a2), R1 = (d1 c1)(d2 c2), J 1 2 3 4 wjr1j {0.0480[0.00878, 0.0427], 0.0996, 0.1786, 0.3282[0.0207, 0.170]} {0.0153[0 .00099, 0.00799], 0.0243, 0.0470, 0.0883[0.0061, 0.0473]} {0.0602[0, 0.0066], 0. 0669, 0.1358, 0.1524[0, 0.0166]} {0.0415, [0.00095, 0.0136], 0.0562, 0.0902, 0.1 216[0.0022, 0.0336]} U1 = {0.165, [0.01072, 0.13119], 0.3159, 0.4516, 0.6905[0.029, 0.2678]} Accordin g to Table 17, the membership function value of summarized as in Table 18. u1 (x) may be

Fuzzy AHP and Its Application Table 18. The Membership Function Value of X 0.165 0.6905 0.3159 x 0.4516 0.165 x 0.3159 0.4516 x 0.6905 0 0 1 [0,1] [0,1] u1(x) 69 u1(x) When x [0.165, 0.3159], it is defined as: x = (0.01072) 2 + (0.13119) + 0.165 and when x [0.4516, 0.6905], it is defined a s: x = (0.029) 2 + (–0.2678) + 0.6905. The fuzzy utilities U2, and U3 can be obtai ned in a similar manner. They are also presented in Figure 2. Figure 2. The fuzzy utilities 2.6 Chang’s (1992) Extent Analysis Method First, the outlines of the extent analysis method on fuzzy AHP are given and the n the method is applied to a catering firm selection problem. Let X x1 , x 2 ,.. ., x n be an object set and U u1 , u 2 ,..., u m be a goal set. According to the method of Chang’s (1992) extent analysis, each object is taken and extent analysi s for each goal, g i , is performed, respectively.

70 T. Demirel et al. Therefore, m extent analysis values for each object can be obtained, with the fo llowing signs: 2 m M 1 , M gi , ..., M gi , gi i 1,2,...,n 1,2,..., m are TFNs. j where all the M gi j The steps of Chang’s extent analysis can be given as in the following: Step 1. The value of fuzzy synthetic extent with respect to ith object is defined as m n m 1 Si j 1 m j i M j gi i 1 j 1 M j gi (18) To obtain j M gi , perform the fuzzy addition operation of m extent analysis values for a particular matrix such that m j M gi j 1 j 1 m m m li , j 1 1 mi , j 1 ui (19) and to obtain n i 1 m j j M gi 1 , perform the fuzzy addition operation of j M gi j 1,2 ,...,m values such that

n m n n n li , i 1 j 1 i 1 i 1 mi , i 1 ui (20) and then compute the inverse of the vector in Eq. (20) such that n m 1 M i 1 j 1 j gi 1 n i , 1 n i , 1 n i (21) l 1 i u 1 i m 1 i

Fuzzy AHP and Its Application 71 Step 2. The degree of possibility of M2 l 2 , m2 , u 2 V M2 M1 M1 y x l1 , m1 , u1 is defined as M1 sup min x, M2 y (22) and can be equivalently expressed as follows: V M2 M1 hgt M 1 M2 1, M2 if m 2 if l1 l1 u 2 u2 m1 l1 m1 u2 (23) d 0, m2 , otherwise where d is the ordinate of highest intersection point D between M 2 (see Figure 3). To compare M1 and V M1 M2 M 1 and M 2 , we need both the values of and V M 2 M 1 . Step 3. The degree of possibility for a convex fuzzy number to be greater i 1, 2 ,..., k can be defined by than k convex fuzzy numbers M i V M M 1 , M 2 ,..., M k V M min V M Mi , M 1 and M i 1,2,..., k M 2 and ...and M Mk

(24) Assume that d Ai min V S i Sk (25)

72 T. Demirel et al. 1 M2 M1 V(M2 M1) l2 m2 l1 d u2 m1 u1 Figure 3. The intersection between M1 and M2 For k 1,2,..., n; k W i . Then the weight vector is given by d A1 , d A2 ,..., d An T (26) where Ai i 1,2,..., n are n elements. Step 4. Via normalization, the normalized weight vectors are W d A1 , d A2 ,..., d An T (27) where W is a nonfuzzy number. In this method, the fuzzy conversion scale is as i n Table 19. A different scale in fuzzy AHP can be found in the literature as in Abdel-Kader and Dugdale’s (2001) study. Table 19. Triangular Fuzzy Conversion Scale Linguistic scale Just equal Equally important Weakly important Strongly more imp ortant Very strong more important Absolutely more important Triangular fuzzy sca le (1, 1, 1) (1/2, 1, 3/2) (1, 3/2, 2) (3/2, 2, 5/2) (2, 5/2, 3) (5/2, 3, 7/2) T riangular fuzzy reciprocal scale (1, 1, 1) (2/3, 1, 2) (1/2, 2/3, 1) (2/5, 1/2, 2/3) (1/3, 2/5, 1/2) (2/7, 1/3, 2/5)

Fuzzy AHP and Its Application 73 2.7 A Numerical Example (Kahraman et al., 2004) A big company wants to contract with a catering firm. Alternative catering firms are Firm1, Firm2, and Firm3. The goal is to select the best catering among the alternatives. The selection hierarchy of the best catering firm is shown in Figu re 4. Figure 4. Selection of the best catering firm The following phases are taken to solve the problem: From Table 20, SH = (3.17, 4.00, 5.00) (1/12.34, 1/10.00, 1/8.14) = (0.26, 0.40, 0.61) (1/12.34, 1/10.00, 1 /8.14) = (0.24, 0.35, SQM = (2.90, 3.50, 4.17) 0.51) SQS = (2.07, 2.50, 3.17) are obtained. (1/12.34, 1/10.00, 1/8.14) = (0.17, 0.21, 0.39) Table 20. The Fuzzy Evaluation Matrix with Respect to the Goal H (1, 1, 1) (2/5, 1/2, 2/3) (2/3, 1, 3/2) QM (3/2, 2, 5/2) (1, 1, 1) (2/5, 1/2, 2/3) QS (2/3,1, 3/2) (3/2, 2, 5/2) (1, 1, 1) H QM QS

74 T. Demirel et al. Using these vectors, V (SH SQM) =1.00, V (SH SQS) =1.00, V (SQM SH) =0.84 V (SQM SQS) =1.00, V (SQS SH) =0.47, and V (SQS SM) = 0.61 are obtained. Thus, the wei ght vector from Table 19 is calculated as WG = (0.43, 0.37, 0.20)T. From Table 2 1, SHM = (0.32, 0.50, 0.74), SHSP = (0.17, 0.25, 0.39), SHSV = (0.17, 0.25, 0.39 ) V (SHM SHSP) = 1.00, V (SHM SHSV) = 1.00, V (SHSP SHM) = 0.21 V (SHSP SHSV) = 1.00, V (SHSV SHM) = 0.21, and V (SHSV SHSP) = 1.00 are obtained and the weight vector from Table 20 is calculated as WH = (0.70, 0.15, 0.15)T. Table 21. Evaluation of the Sub-Attributes with Respect to Hygiene (H) HM (1, 1, 1) (2/5, 1/2, 2/3) (2/5, 1/2, 2/3) HSP (3/2, 2, 5/2) (1, 1, 1) (2/3, 1 , 3/2) HSV (3/2, 2, 5/2) (2/3, 1, 3/2) (1, 1, 1) HM HSP HSV The weight vector from Table 22 is calculated as WQM = (0.19, 0.04, 0.77, 0.00)T . Table 22. Evaluation of the Sub-Attributes with Respect to Quality of Meal (QM) VM (1, 1, 1) (2/5, 1/2, 2/3) (5/2, 3, 7/2) (2/7, 1/3, 2/5) CoM (3/2, 2, 5/2) (1, 1, 1) (5/2, 3, 7/2) (2/9, 1/4, 2/7) CaM (2/7, 1/3, 2/5) (2/7, 1/3, 2/5) (1, 1, 1) (2/7, 1/3, 2/5) TM (5/2, 3, 7/2) (7/2, 4, 9/2) (5/2, 3, 7/2) (1, 1, 1) VM CoM CaM TM The weight vector from Table 23 is calculated as WQS = (0.00, 0.05, 0.00, 0.95)T . Table 23. Evaluation of the Sub-Attributes with Respect to Quality of Service (Q S) BSP (1, 1, 1) (5/2, 3, 7/2) (2/9, 1/4, 2/7) (7/2, 4, 9/2) ST (2/7, 1/3, 2/5) (1, 1, 1) (2/7, 1/3, 2/5) (5/2, 3, 7/2) CP (7/2, 4, 9/2) (5/2, 3, 7/2) (1, 1, 1) (7 /2, 4, 9/2) PS (2/9, 1/4, 2/7) (2/7, 1/3, 2/5) (2/9, 1/4, 2/7) (1, 1, 1) BSP ST CP PS The weight vector from Table 24 is calculated as WHM = (0.66, 0.00, 0.34)T.

Fuzzy AHP and Its Application Table 24. Evaluation of the Catering Firms with Respect to Hygiene of Meal (HM) Firm1 (1, 1, 1) (2/7, 1/3, 2/5) (2/5, 1/2, 2/3) Firm2 (5/2, 3, 7/2) (1, 1, 1) (5 /2, 3, 7/2) Firm3 (3/2, 2, 5/2) (2/7, 1/3, 2/5) (1, 1, 1) 75 Firm1 Firm2 Firm3 The weight vector from Table 25 is calculated as WHSP = (0, 0, 1)T. Table 25. Evaluation of the Catering Firms with Respect to Hygiene of Service Pe rsonnel (HSP) Firm1 (1, 1, 1) (2/3, 1, 3/2 ) (7/2, 4, 9/2) Firm2 (2/3, 1, 3/2) (1, 1, 1) (3/2, 2, 5/2) Firm3 (2/9, 1/4, 2/7) (2/5, 1/2, 2/3) (1, 1, 1) Firm1 Firm2 Firm3 The weight vector from Table 26 is calculated as WHSV = (0, 0, 1)T. Table 26. Evaluation of the Catering Firms with Respect to Hygiene of Service Ve hicles (HSV) Firm1 (1, 1, 1) (2/3, 1, 3/2) (5/2, 3, 7/2) Firm2 (2/3,1, 3/2) (1, 1, 1) (3/2, 2 , 5/2) Firm3 (2/7, 1/3, 2/5) (2/5, 1/2, 2/3) (1, 1, 1) Firm1 Firm2 Firm3 The weight vector from Table 27 is calculated as WVM = (0, 0, 1)T. Table 27. Evaluation of the Catering Firms with Respect to Variety of Meal (VM) Firm1 (1, 1, 1) (5/2, 3, 7/2) (2/3, 1, 3/2) Firm2 (2/7, 1/3, 2/5) (1, 1, 1) (1, 1, 1) Firm3 (2/3, 1, 3/2) (1, 1, 1) (1, 1, 1) Firm1 Firm2 Firm3 The weight vector from Table 28 is calculated as WCoM = (0.87, 0.00, 0.13)T. Table 28. Evaluation of the Catering Firms with Respect to Complementary Meals i n a Day (CoM) Firm1 (1, 1, 1) (2/7, 1/3, 2/5) (2/3, 1, 3/2) Firm2 (5/2, 3, 7/2) (1, 1, 1) (1, 1, 1) Firm3 (2/3, 1, 3/2) (1, 1, 1) (1, 1, 1) Firm1 Firm2 Firm3

76 T. Demirel et al. Using the similar calculations, the weight vectors of the catering firms with re spect to Calorie of meal (CaM) is obtained as WCaM = (0.00, 0.31, 0.69)T. Taste of meal (TM) is obtained as WTM = (0.27, 0.18, 0.55)T. Behavior of service perso nnel (BSP) is obtained as WBSP = (1, 0, 0)T. Service time (ST) is obtained as WS T = (0.05, 0.64, 0.31)T. Communication on phone (CP) is obtained as WCP = (0.72, 0.00, 0.28)T. Problem solving ability (PS) is obtained as WPS = (0, 0, 1)T. Table 29. Obtained Results Sub-attributes of hygiene HM HSP HSV Alternative priority weight Weight 0.70 Alternative Firm1 0.66 Firm2 0 Firm3 0.34 Sub-attributes of quality of meal VM CoM Weight 0.19 0.04 Alternative Firm1 0 0.87 Firm2 0 0 Firm3 1 0.13 Sub-attributes of quality of service BSP ST Weight 0.00 0.05 Alternative Firm1 1 0.05 Firm2 0 0.64 Firm3 0 0.31 Main attributes of the goal H Weight 0.43 Altern ative Firm1 0.46 Firm2 0 Firm3 0.54 0.15 0 0 1 CaM 0.77 0 0.31 0.69 CP 0.00 0.72 0 0.28 QM 0.37 0.03 0.24 0.73 0.15 0 0 1 TM 0.00 0.27 0.18 0.55 PS 0.95 0 0 1 QS 0.20 0.003 0.032 0.965 0.21 0 .10 0.69 0.003 0.032 0.965 0.03 0.24 0.73 0.46 0.00 0.54 The combination of priority weights for sub-attributes, attributes, and alternat ives to determine priority weights for the best catering firm are

Fuzzy AHP and Its Application 77 shown in Table 29. With respect to the results, firm3 is the catering firm selec ted. 2.8 Cheng’s (1996) Entropy-Based Fuzzy AHP The Shannon entropy, H, which is applicable only to probability measures, assume s the following form in evidence theory (Klir and Yan, 1995): n H m j 1 m x log 2 m x . (28) This function, which forms the basis of classic information theory, measures the average uncertainty associated with the prediction of outcomes in a random expe riment. Its range is 0, log 2 X . Clearly, H m when m x 1 for some x 0. X; H m log 2 X when m defines the 1/ X , x X . uniform probabilities distribution on X i.e., m x The principle of maximum uncertainty is well developed and broadly utilized with in classic information theory, where it is called the principle of maximum entro py. Cheng’s [1996] evaluation model can be described as given below: Step 1. Construct a hierarchy structure for any problem. Step 2. Build membershi p function of judgment criteria. Step 3. Compute the performance score. Step 4. Utilize fuzzy AHP method and entropy concepts to calculate aggregate weights. Th e computational procedure of this decision-making methodology is summarized as f ollows. To compare the performance scores, we can use symmetric triangular fuzzy numbers 1 , 3 , 5 , 7 , 9 to indicate the relative strength of the element s in the hierarchy matrix. ~ ~ ~ ~ ~

78 T. Demirel et al. ~ To assemble the total fuzzy judgement matrix A , we can multiply the ~ fuzzy s ubjective weight vector W with the corresponding column of fuzzy ~ judgement mat rix X . Thus, we get ~ A ~ w1 ~ w 1 ~ x11 ~ x 21 ~ w2 ~ w 2 ~ x12 ~ x 22 ~ wn ~ w n ~ x1n ~ x 2n . (29) ~ w1 ~ xn1 ~ w2 ~ xn 2 ~ wn ~ xnn Now fuzzy number multiplications and additions using the interval arithmetic and cuts are made, and Eq. (30) is obtained. ~ A a11l , a11u a n1l , a n1u a1nl , a1nu (30) a nn1 , a nnu 1 and all i, j.

where a ijl wil xijl , aiju wiu xiju , for 0 < Now the degree of satisfaction of the judgment  will be estimated. When is fixed, we will set the index of optimism by the degree of the optimism of a decision m aker. A larger indicates a higher degree of optimism. The index of optimism is a linear convex combination it is explained by ˆ aij Thus we have 1 aijl aiju , 0, 1 . (31) ˆ A ˆ a11 ˆ a 21 ˆ a n1 ˆ a12 ˆ a 22 ˆ an2 ˆ a1n ˆ a 2n ˆ a nn (32) ˆ where A is a precise judgment matrix.

Fuzzy AHP and Its Application 79 The entropy must be first calculated by using the relative frequency of Eq. (33) and the entropy formula of Eq. (34), i.e., a11 s1 a n1 sn where a12 s1 an2 sn a1n s1 a nn sn f 11 f n1 f 12 f n2 f 1n (33) f nn sk n j 1 a kj . We can use this equation to caloculate the entropy, i.e., n H1 j 1 n f1 j log 2 f1 j f 2 j log 2 f 2 j j 1 n H2 Hn j 1 (34) f nj log 2 f nj where H i is ith entropy value. The entropy weights can be determined by using E q. (35). Hi Hi n , i 1, 2,..., n (35) Hj

j 1 2.9 A Numerical Example A company wants to choose one supplier among four suppliers. They determine five attributes: capacity, quality, cost, distance, and delivery time. By the help o f the experts, they determine all the suppliers, and they are given in Table 30. Also for the company they give fuzzy weights of the criteria and they are given in Table 31. This work is done for choosing the best supplier for the company.

80 Table 30. Fuzzy Judgment Matrix X1 (1, 3, 5) (7, 9, 9) (1, 1, 3) (3, 5, 7) X2 (5, 7, 9) (1, 1, 3) (1, 3, 5) (5, 7, 9) X3 (3, 5, 7) (1, 3, 5) (1, 3, 5) (1, 1, 3) X4 (1, 3, 5) (1, 3, 5) (3, 5, 7 ) (1, 1, 3) T. Demirel et al. A1 A2 A3 A4 X5 (1, 1, 3) (3, 5, 7) (1, 3, 5) (1, 3, 5) Table 31. Fuzzy Subjective Weight Vector X1 (1, 3, 5) X2 (7, 9, 9) X3 (5, 7, 9) X4 (1, 3, 5) X5 (1, 1, 3) W The following phases are taken to solve the problem: ~ To assemble the total fuzzy judgment matrix A , we can multiply the ~ fuzzy su bjective weight vector W by the corresponding column. ~ A X1 A1 A2 A3 A4 (1, 3, 5) (1, 3, 5) (1, 3, 5) (1, 3, 5) (1, 3, 5) (7, 9, 9) (1, 1 , 3) (3, 5, 7) X2 (7, 9, 9) (7, 9, 9) (7, 9, 9) (7, 9, 9) X3 (5, 7, 9) (5, 7, 9) (1, 1, 3) (5, 7, 9) (1, 3, 5) (5, 7, 9) (5, 7, 9) (5, 7, 9) X4 (3, 5, 7) (1, 3, 5) (1, 3, 5) (1, 3, 5) (1, 3, 5) (1, 3, 5) (1, 1, 3) (1, 3, 5) X5 (1, 3, 5) (1, 1, 3) (1, 3, 5) (1, 1, 3) (3, 5, 7) (1, 1, 3) (1, 1, 3) (1, 1, 3) (1, 1, 3) (3, 5, 7) (1, 3, 5) (1, 3, 5) 0, 1 , ~ A aL , a R a2 a1 a1 , a3 a2 a3 (5 3) Set = 0.8 and = 0.5 for a moderate decision maker. â11 = [(3 1) × 0.8 + 1, (5 3) × 0.8 + 5] [(3 1) × 0.8 + 1, × 0.8 + 5] â11 = [6.76, 11.56] All the results are given below . ~ A 0.8 X1 A1 A2 A3 A4 [6.76, 11.56] [22.36, 30.6] [2.6, 4.76] [11.96, 18.36] X2

[56.76, 66.6] [8.6, 12.6] [22.36, 30.6] [56.76, 66.6] X3 [30.36, 39.96] [17.16, 25.16] [17.16, 25.16] [6.6, 10.36] X4 [6.76, 11.56] [6.76, 11.56] [11.96, 18.36] [2.6, 4.76] X5 [1, 1.96] [4.6, 7.56] [2.6, 4.76] [2.6, 4.76]

Fuzzy AHP and Its Application 81 where = 0.8. ˆ We compute to a11 by using Eq. (31) as ˆ a11 1 0.5 6.76 0.5 11.56 9.16 All the results are given below. ˆ A A1 A2 A3 A4 X1 9.16 26.48 3.68 15.16 X2 61.68 10.6 26.48 61.68 X3 35.16 21.16 21.16 8.48 X4 9.16 9.16 15.16 3.68 X5 1.48 6.08 3.68 3.68 where = 0.5. We calculate relative frequencies by Eq. (33). f A1 A2 A3 A4 X1 0.0785 0.3603 0.0524 0.1635 X2 0.5288 0.1442 0.3774 0.6655 X3 0.3 014 0.2879 0.3015 0.0914 X4 0.0785 0.1246 0.2160 0.0397 X5 0.0126 0.0827 0.0524 0.0397 Then, we compute entropy values by using the relative frequencies and the entrop y formula Eq. (34). The resultant aggregate weights can be determined by normali zing entropy values. Entropy Value H1 = 1.6635 H2 = 2.1225 H3 = 1.9754 H4 = 1.5053 Entropy Weight 0.2 289 0.2921 0.2719 0.2069 A1 A2 A3 A4 From the last table, supplier A2 is the best choice when = 0.5. = 0.8 and 3. CONCLUSION Decisions are made today in increasingly complex environments. The fuzzy AHP pro vides a systematic method for comparison and weighting of the multiple criteria and alternatives to decision makers in the case of incomplete information. Many alternative fuzzy AHP methods exist in the

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A SWOT-AHP APPLICATION USING FUZZY CONCEPT: E-GOVERNMENT IN TURKEY Cengiz Kahraman1, Nihan Çetin Demirel2, Tufan Demirel2, and Nüfer Yasin Ate 1 1 Department of Industrial Engineering, Istanbul Technical University, Macka, Ista nbul, Turkey 2Department of Industrial Engineering, Yildiz Technical University, Yildiz, Istanbul, Turkey Abstract: E-government refers to the delivery of information and services online via the I nternet. Many governmental units across the world have embraced the digital revo lution and placed a wide range of materials on the web, from publications to dat abases. The purpose of this study is to evaluate and to determine the alternativ e strategies for e-government applications in Turkey. We use the strengths, weak nesses, opportunities, and threats (SWOT) approach in combination with the crisp and fuzzy analytic hierarchy process (AHP) to achieve this task. The strategies have been prioritized by using both methods comparatively and sensitivity analy ses of the obtained results have been presented. Outranking, fuzzy outranking re lation, pair-wise comparison, e-government, SWOT, analytic hierarchy process, st rategic planning, sensitivity analysis Key words: 1. INTRODUCTION Digital technologies serve as a basic source of transformation in economies, com munities, and government functions all over the world. The occurrence of technol ogical change of the late 1990s, the result of which was the enabling of the del ivery of services over the internet, caused major and rapid transformation of ho w governments function. Development of e-commerce and the evolution projected fo r the near future has encouraged consumers to demand more and more customized, r apid, and C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 85

86 C. Kahraman et al. at-home services. The rapid development of modern information and communication technologies is having far-reaching effects on all aspects of modern life, inclu ding government. Academics have suggested various definitions for e-government. According to Kaylor et al. (2001), e-government is taken to be the ability for c itizens to communicate and/or interact with the city via the Internet in any way more sophisticated than a simple e-mail letter to the generic city (or webmaste r) or e-mail address provided at the site. The United Nations and the American S ociety for Public Administration (2002) defined egovernment as “utilizing the inte rnet and the World-Wide-Web for delivering government information and services t o citizens.” More recently, e-government is defined by the OECD (2003) as “the use o f Information and Communications Technologies (ICT), and particularly the Intern et, as a tool to achieve better government.” Many works on e-government have been published. Most of these works are on strategy evaluation, future development pr ograms, and scenario planning. Layne and Lee (2001) described different stages o f egovernment development. The stages of development outline the structural tran sformations of government as they progress toward electronically enabled governm ent. And they also described how the Internet-based government emerged with trad itional models amalgamated with traditional public administration, implying fund amental changes in the form of government. They developed a four-stage growth mo del with themselves providing observation. Chen and Gant (2001) examined the pot ential of application service providers to transform electronic government servi ces at the local level. Gupta and Jana (2003) suggested a flexible framework to choose an appropriate strategy to measure the tangible and intangible benefits o f e-government. Reddick (2004) explored the current stages of development and pr ospect for future development in e-government growth in the U.S. cities. Akman e t al. (2005) reviewed and discussed e-government issues in general, its global p erspective, and then reported the findings of a survey concerning impact of gend er and education among the e-government users in Turkey. Gil-Garcia and Pardo (2 005) examined the extent to which information systems (IS) research informs the development of practitioner tools for government information technology (IT) dec ision makers. SWOT, the acronym standing for strengths, weaknesses, opportunitie s and threats analysis, is a commonly used tool for analyzing internal and exter nal environments to attain a systematic approach and support for a

A SWOT-AHP Application using Fuzzy Concept 87 decision situation. The internal and external factors most important to the ente rprise’s future are referred to as strategic factors, and they are summarized with in the SWOT analysis. The final goal of a strategic planning process, of which S WOT is an early stage, is to develop and adopt a strategy resulting in a good fi t between internal and external factors. SWOT can also be used when strategy alt ernative emerges suddenly and when the decision context relevant to it has to be analyzed. This chapter proposes a multi-attribute decision-making-based SWOT an alysis for the evaluation of alternative e-government strategies for Turkey. Aft er a wide literature review, it is found that the application of this methodolog y to e-government area is described for the first time in this chapter. In this chapter, we applied a SWOT analysis using the crisp and the fuzzy approaches of a multi-attribute evaluation method that is called the analytic hierarchy proces s (AHP) to the e-government process of Turkey. E-government strategy selection w ith SWOT analysis is a complex problem in which many qualitative aspects must be considered. These kinds of aspects make the evaluation process hard and vague. The judgments from experts are always vague and linguistic rather than exact val ues. Thus, it is suitable and flexible to express the judgments of experts in fu zzy quantities instead of in crisp quantities. Additionally, the hierarchical st ructure is a good approach to describe these kinds of complicated evaluation pro blems. Fuzzy AHP has the capability of taking these situations into account with a hierarchical structure. To be able to compare with the crisp case, we also im plemented the crisp AHP. We first determined the factors in the SWOT groups and alternatives strategies for e-government application in Turkey. Then we computed the importance weights of these factors and the scores of the strategies. The a im of this study is to determine the priorities of the e-government strategies f or the case of Turkey. The remainder of this chapter is organized as follows. Se ction 2 presents Turkey’s position among the other countries and e-government proj ects and services in Turkey. Section 3 introduces some terminology from SWOT ana lysis and the analytic hierarchy process (crisp and fuzzy cases). The method for utilizing AHP in SWOT analysis is also defined in the third section. SWOT analy sis for e-government in Turkey is presented in Section 4. Sections 5 and 6 defin e possible e-government strategies and the evaluation of e-government strategies in Turkey. The last section summarizes the findings and makes suggestions for f urther research.

88 C. Kahraman et al. 2. 2.1 E-GOVERNMENT IN TURKEY Turkey’s Position Among the Other Countries The e-government agenda is being pursued throughout the world to one degree or a nother, but it has added significance in Central Europe. The region is just begi nning to emerge from a period of far-reaching political and economic transformat ion after the collapse of repressive communist systems. For these countries, e-g overnment is more than simply a new channel of delivering services; it offers an opportunity to achieve a quantum leap in transparency and efficiency of adminis tration, which the region’s leaders have promised their citizens since the early 1 990s. In order to gauge their capacity to implement such change as well as their progress to date, the Economist Intelligence Unit (EIU), sponsored by Oracle, c onducted a wideranging analysis of the e-government experience in the Central Eu rope region. EUI considered seven criteria with different weightings: connectivi ty and technology infrastructure (CTI) (20% weight), business and legal environm ent (BLE) (10% weight), e-democracy (E-D) (15%), education and skills (ES) (10%) , online public services for citizens (OPSC) (15%), online public service for bu sinesses (OPSB) (15%), and government policy and vision (GPV) (15%). Table 1 exp resses the results of this analysis in comparative fashion. Scores are on a scal e of 1 to 10 (Source: Economist Intelligence Unit). The rankings cover the ten n ew and candidate EU members from Central Europe, as well as another prospective member, Turkey. Table 1. Economist Intelligence Unit Central Europe e-government rankings Overal l score Category weight Estonia Czech Rep. Slovenia Poland Hungary Turkey Lithua nia Latvia Slovakia Romania Bulgaria 5.87 5.67 5.33 4.74 4.69 4.64 4.62 4.58 4.4 4 3.99 3.71 CTI 0.20 3.37 3.98 3.68 2.43 3.15 2.67 2.21 2.34 2.80 1.43 1.92 BLE 0.10 6.80 6.95 6.60 6.60 6.66 4.23 6.36 6.32 6.28 5.42 5.50 ES 0.10 7.67 7.33 7. 33 6.67 7.00 5.67 6.33 6.67 6.67 5.33 5.67 GPV 0.15 6.50 6.10 5.00 5.30 5.50 4.9 0 4.70 5.00 3.80 4.70 3.10 E-D 0.15 4.60 3.60 2.90 2.90 3.30 4.20 2.60 2.60 2.90 2.60 2.60 OPSC 0.15 6.38 5.68 6.73 5.98 5.00 5.70 5.00 4.79 4.46 4.08 3.95 OPSB 0.15 7.52 7.57 6.68 5.33 4.19 6.00 7.08 6.35 6.08 6.16 5.08

A SWOT-AHP Application using Fuzzy Concept 89 2.2 Process of Turkey’s Development in e-Government Since the beginning of the 1990s, there has been an increase in the effort by mo st countries to transform into an information society. Essentially, economic and social necessities bring about these efforts. The Turkish Government initiated the Urgent Action Plan in December 2002 to remedy long-lasting economic problems and to improve the social well-being of the country. One basic component of thi s plan is the “e-Transformation Turkey Project,” which aims to turn Turkey into an i nformation society. Some objectives of this project are to facilitate the partic ipation of citizens to the decision-making process; to enhance transparency and accountability for the public management; to promote ICT diffusion; and to coord inate egovernment investments by means of information and communication technolo gies. The Minister of State and Deputy Prime Minister has the high level respons ibility of the project, and the project is coordinated by the State Planning Org anization (SPO). In order to realize the objectives of this project and to ensur e the success of the project, a new coordination unit, the Information Society D epartment, within SPO was established. This department is responsible for the ov erall coordination of the project. Before this project was launched, lack of eff icient coordination between institutions made the progress slow and ineffective. For the first time in Turkey, a dedicated department, which is believed to be a crucial element for success, has been named as the coordinator of information s ociety activities. To increase the participation and the level of success, an Ad visory Committee with 41 members has been established. This consulting body cons ists of the representatives of public institutions, nonprofit organizations, and universities (Akman et al., 2005) In line with the government’s schedule, the ini tial focal point in this project was the Short Term Action Plan (STAP), which co vered 2003 2004, for implementing specific tasks. The first action of STAP was t he determination of an “Information Society Strategy,” which encompassed every part of society and maximized national benefits and value added. As in the preparatio n phase, the implementation of STAP and all other related activities was coordin ated by the SPO-Information Society Department and was open to every contributio n in order to successfully achieve the ultimate goal: to transform Turkey into a n information society. The e-Transformation Turkey Executive Board was also esta blished with the same circular that validates STAP. The Board is composed of the Minister of State and Deputy Prime Minister (e-minister), Minister of

90 C. Kahraman et al. Industry and Trade, Minister of Transport, Undersecretary of SPO, and Chief Advi sor to the Prime Minister. The Board was given the responsibility of supervision of the e-Transformation Turkey Project. 2.3 E-Government Projects and Services In Turkey Akman et al. (2002) classify and report a list of current e-government projects. A classification based on the project characteristics is given as follows: Proj ects related to national IS and services; projects related to education, culture , youth, and sports; projects related to health, family, labor, and social affai rs; projects related to finance and economics; projects related to interior affa irs; projects related to justice affairs; projects related to agriculture, fores try, village, and environmental affairs; projects related to industry, technolog y, energy, and natural resources; projects related to communications, public wor k, tourism, and development and housing; and projects related to foreign affairs . From the details of the report, one can observe that most of the projects are for processing information and hence devoted to services. However, a considerabl e number of these systems is being developed for government-to-government (G2G) communications. Although these systems do not adopt the government-to-citizens ( G2C) approach entirely, the above classification provides evidence that these wi ll constitute a sufficient base for G2C communication in the future in almost al l areas of public affairs. Currently available e-government services in Turkey a re classified into three groups. G2G services are the ones enjoyed among public organizations electronically. G2C services are given by government organizations to citizens electronically. Governmentto-business (G2B) services are given by g overnment organizations to private sector. 3. SWOT-AHP ANALYSIS FOR E-GOVERNMENT In the following discussion, the fundamentals of SWOT analysis and AHP are given . Later, these techniques are combined to prioritize the e-government strategies . 3.1 SWOT Analysis A scan of the internal and external environment is an important part of the stra tegic planning process. Environmental factors internal to the

A SWOT-AHP Application using Fuzzy Concept 91 organization usually can be classified as strengths (S) or weaknesses (W), and t hose external to the organization can be classified as opportunities (O) or thre ats (T). Such an analysis of the strategic environment is called to as a SWOT an alysis. The SWOT approach involves systematic thinking and comprehensive diagnos is of factors relating to a new product, technology, management, or planning (We ihrich, 1982). Figure 1 shows how SWOT analysis fits into an environment scan. Environment Scan Internal Analysis External Analysis Strengths Weaknesses Opportunities Threats SWOT Matrix Figure 1. SWOT analysis framework 3.2 A Multi-Attribute Evaluation Method: AHP The analytic hierarchy process has been used in many different fields as a multi -attribute decision analysis tool with multiple alternatives and criteria. An ex tensive literature review on AHP can be found in Vaidya and Kumar’s (2006) study. AHP uses “pair-wise comparisons” and matrix algebra to weigh criteria. The decision is made by using the derived weights of the evaluative criteria (Saaty, 1980). A fter the hierarchy of the problem is constructed, the matrices of pairwise compa risons are obtained. In this matrix, the element aij = 1/aij, and thus, when i = j, aij = 1. The value of wi may vary from 1 to 9, and 1/1 indicates equal impor tance, whereas 9/1 indicates extreme or absolute importance. The scale is shown in the Table 2. Table 2. Evaluation Scale Num Value 1 3 5 7 9 2, 4, 6, 8 Verbal Scale Equal impo rtance Moderate importance Strong importance Very strong importance Extreme or a bsolute importance Intermediate values

92 C. Kahraman et al. 1 A (aij ) w2 / w1 wn / w1 w1 / w2 1 wn / w2 w1 / wn w2 / wn 1 (1) In the comparisons, some inconsistencies can be expected and accepted. When A co ntains inconsistencies, the estimated priorities can be obtained by using the A matrix as the input using the eigenvalue technique. (A max I )q 0 (2) where max is the largest eigenfactor of matrix A of size n, q is its correct eig enfactor and I is the identity matrix of size n. The correct eigenfactor, q, con stitutes the estimation of relative priorities. Each eigenfactor is scaled to su m up to one to obtain the priorities. Saaty (1977) demonstrated that max = n is a necessary and sufficient condition for consistency. Inconsistency may occur wh en max deviates from n due to inconsistent responses in pair-wise comparisons. T herefore, the matrix A should be tested for consistency using index, CI, which h as been constructed. CI ( max n ) /( n 1 ) (3) CI estimates the level of consistency with respect to a comparison matrix. Then, because CI is dependent on n, a consistency ratio CR is calculated, which is de pendent of n as shown below. CR CI / RI (4) where CI is the consistency index, RI is random index (RI) generated for a rando m matrix of order n, and CR is the consistency ratio (Saaty, 1993). The general rule is that CR 0.1 should be maintained for the matrix to be consistent. Otherw ise, all or some comparisons must be repeated in order to resolve the inconsiste ncies of the pair-wise comparisons. 3.3

The Fuzzy AHP To deal with vagueness of human thought, Zadeh (1965) first introduced the fuzzy set theory, which was oriented to the rationality of uncertainty

A SWOT-AHP Application using Fuzzy Concept 93 due to imprecision or vagueness. A major contribution of fuzzy set theory is its capability of representing vague data. The theory also allows mathematical oper ators and programming to apply to the fuzzy domain. A fuzzy set is a class of ob jects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function, which assigns to each object a grade of m embership ranging between zero and one. A tilde “ ” will be placed above a symbol if the symbol represents a ~ fuzzy set. A triangular fuzzy number (TFN), M is show n in Figure 2. A TFN is denoted simply as (l / m, m / u ) or ( l , m, u ). The p arameters l, m, and u respectively, denote the smallest possible value, the most promising value, and the largest possible value that describe a fuzzy event. 1.0 Ml(y) Mr(y) 0.0 l m u Figure 2. A Triangular Fuzzy Number, M ~ Each TFN has linear representations on its left and right side such that its mem bership function can be defined as x M 0, ( x l ) /(m l ), (u x) /(u m), x l l x x l m (5) A fuzzy number can always be given by its corresponding left and right represent ation of each degree of membership: ~ M (M l ( y) , M r ( y) l m l y, u m u y y 0, 1 (6)

94 C. Kahraman et al. where l(y) and r(y) denote the left side representation and the right side repre sentation of a fuzzy number, respectively. Many ranking methods for fuzzy number s have been developed in the literature. These methods may give different rankin g results, and most methods are tedious in graphic manipulation requiring comple x mathematical calculation. The algebraic operations with fuzzy numbers can be f ound in Zimmermann (1994). A basic literature review on fuzzy AHP can be found i n Kahraman et al.’s (2004) study. In this chapter, we prefer Chang’s (1992, 1996) ex tent analysis method since the steps of this approach are relatively easier than the other fuzzy AHP approaches and similar to the conventional AHP. This method can be found with its details in Chapter 3, Section 3.2.6. 3.4 The Method for Using AHP in SWOT Analysis The idea in using AHP within a SWOT framework is to systematically evaluate SWOT factors and commensurate their intensities. If it is used in combination with t he analytic hierarcy process, the SWOT approach can provide a quantitative measu re of importance of each factor on decision making (Saaty and Vargas, 2001). The method introduced proceeds as follows (Kurttila et al., 2000): Step 1. SWOT ana lysis is carried out. The relevant factors of the external and internal environm ents are identified and included in SWOT analysis. When standard AHP is applied, it is recommended that the number of factors within a SWOT group should not exc eed 10 because the number of pair-wise comparisons needed in the analysis increa ses rapidly (Saaty, 1980). Thus, the result of the comparisons is quantitative v alues expressing the priorities of the factors included in SWOT analysis. Step 2 . Pair-wise comparisons between SWOT factors are carried out within every SWOT g roup. When making the comparisons, the questions at stake are as follows: (1) wh ich of the two factors compared is a greater strength (opportunity, weakness, or threat); and (2) how much greater. With these comparisons as the input, the rel ative local priorities of the factors are computed using the eigenvalue method. These priorities reflect the decision maker’s perception of the relative importanc e of the factors. Step 3. Pair-wise comparisons are made among the four SWOT gro ups.

A SWOT-AHP Application using Fuzzy Concept 95 The factor with the highest local priority is chosen from each group to present the group. These four factors are then compared as in Step 2. These are the scal ing factors of the four SWOT groups, and they are used to calculate the global p riorities of the independent factors within them. This is done by multiplying th e factors’ local priorities (defined in Step 2) by the value of the corresponding scaling factor of the SWOT group. The global priorities of all factors sum up to one. Step 4. The results are used in the strategy formulation and evaluation pr ocess. The contribution to the strategic planning process comes in the form of n umerical values for the factors. New goals may be set, strategies may be defined , and such implementations may be planned as take into close consideration the f oremost factors. 4. SWOT ANALYSIS FOR E-GOVERNMENT IN TURKEY In the following discussion, we determine the subfactors of the strengths, weakn esses, opportunities, and threats for e-government in Turkey. These subfactors a re used in the prioritization of the e-government strategies. 4.1 Strengths The three main strengths are determined as follows. 4.1.1 Formation of Superviso ry and Executive Committees The Prime Minister of Turkey made a declaration that was published in the Octobe r 4, 2003, issue of the Official Gazette for the realization of the STAP coverin g the years 2003 and 2004 for the e-Transformation project. The declaration spec ified the tasks along with their priorities, the formation of the supervisory an d executive committees, and the responsible organizations in charge of implement ation. The Supervisory committee is headed by the Deputy Prime Minister and incl udes members from top level management of the public and private sectors and the NGO representatives. The members of the Executive Committee are Deputy Prime Mi nister (Chair), Minister of Industry and Trade, Minister of Transportation, Unde rsecretary of the SPO, and Head Advisor of the Prime

96 C. Kahraman et al. Minister. The representatives of nongovernmental organizations, Trade Chambers U nion, Informatics Association of Turkey, Turkish Informatics Foundation, The Ass ociation of Turkish IT Industrialists, the head of the Telecommunications Author ity, and the general manager of Turkish Telecom are also members of this Committ ee. Some of the major achievements realized by the committee have been the settl ement of three laws, namely, the e-Signature law, Knowledge Acquisition law, and Privacy law (Akman et al., 2005). 4.1.2 e-Transformation Projects The Turkish Government has started many e-government projects as indicated in Se ction 4.2.4. These projects are G2G, G2C, and G2B services. They are the main lo comotives of e-government in Turkey. 4.1.3 Support from Top–Level Management of th e Public and Private Sectors Both public and private sectors support the e-government projects. Many minister s from the government and many associations and foundations are the members of t he e-government committees. 4.2 Weaknesses The four main weaknesses are determined as follows. 4.2.1 Lack of Access to Inte rnet Among Certain Sections of The Population It is a particular problem for public sector organizations, as they cannot choos e their customers. Indeed many public services are provided specifically for vul nerable or low-income groups who are the least likely to have access to the tech nology. The main consequence is that public sector organizations will have to co ntinue to provide services through multiple channels at least in the short term to prevent excluding those who do not have access to the Internet (Akman et al., 2005).

A SWOT-AHP Application using Fuzzy Concept 97 4.2.2 Lack of Finance for Capital Investment in New Technologies The reason is that IT was often not viewed as a priority when competing for scar e resources against other claims for capital investment, for example, new school s, roads, and so on. 4.2.3 Need To Change Individual Attitudes and Organizationa l Cultures It is part of an organizational change issue. Another problem is with security a nd authentication that prevented the development of electronic transaction servi ces. It is a specific problem with public sector organizations as the public gen erally saw them as being in a position of trust. 4.2.4 Poor Economic Power of Ci tizens and Businesses Although gross national product per person was US$3,412 in 2003, it was US$4,172 in 2004. This increase means US$348 per month for one person. It is clear that citizens having this level of economic power cannot buy a computer and other sof tware and hardware. 4.3 Opportunities Nine subfactors of opportunities for e-government in Turkey are determined. The first three subfactors are selected by the experts for the SWOT analysis since t hese are evaluated as the most important issues of e-government in Turkey. 4.3.1 A Candidate Country from the European Union’s Information Society Perspective Turkey has been a candidate country according to the European Union (EU) for a l ong time. Being a member of the EU will force Turkey to implement e-government c onditions. So we accept it as an opportunity for Turkey. 4.3.2 Efficiency As with many information technology-related projects, one of the anticipated ben efits is improved efficiency. In e-government projects, this

98 C. Kahraman et al. efficiency can take many forms. Some projects seek to reduce errors and improve consistency of outcomes by automating standardized tasks. A related efficiency g oal of many e-government initiatives is to reduce costs and layers of organizati onal processes by re-engineering and streamlining operating procedures. 4.3.3 Ne w and Improved Services Another opportunity promoted by e-government supporters is the potential to impr ove the quality, range, and accessibility of services. Some observers suggest th at, in addition to enhanced efficiency, the quality of services may be improved through quicker transactions, improved accountability, and better processes. The evolution of e-government also creates the potential for new services. 4.3.4 In creased Citizen Participation A third benefit anticipated by some e-government advocates is increased citizen participation in government. One way this could occur is by connecting people wh o live in remote areas of the country so that they can send and receive informat ion more easily. A second way suggested by some observers is through increased p articipation in government by younger adults. 4.3.5 Improved National Informatio n Infrastructure A fourth possible benefit of the drive to implement e-government initiatives is the improvement of the national information infrastructure. 4.3.6 Potential Chal lenges to e-Government On the other hand, despite the potential opportunities for the implementation of e-government initiatives, several challenges that could prevent the realization of these anticipated benefits. Some of the challenges, such as disparities in c omputer access (digital divide—the lack of equal access to computers, whether due to a lack of financial resources or necessary skills), are preexisting condition s that are connected to larger issues.

A SWOT-AHP Application using Fuzzy Concept 99 4.3.7 Privacy Related to computer security, privacy also presents a challenge to the implement ation and acceptance of e-government initiatives. Concerns about the use of “cooki es,” sharing information between agencies (computer matching), and the disclosure or mishandling of private information are frequent subjects of debate. Addressin g the issue of privacy in the context of e-government may require both technical and policy responses. 4.3.8 Disparities in Computer Access Another challenge for e-government are disparities in computer access. This chal lenge includes two policy issues: the often described “digital divide” and accessibi lity for people with disabilities. In the case of the digital divide, not all ci tizens currently have equal access to computers, whether due to a lack of financ ial resources or necessary skills. Although the placement of Internet-enabled co mputers in schools and public libraries is helping address this issue, these eff orts are still progressing. 4.3.9 Government Information Technology Management a nd Funding A multilayered challenge for the development of e-government is government infor mation technology management and funding, which includes issues such as governme nt information technology worker recruitment, retention, and compensation and co operation between local, state, and federal governments. 4.4 Threats The four main threats are determined as follows. A. Decentralized Internet Gover nance Various bodies and companies assert control over parts of the Internet and try t o exercise it through technical and political means. This threatens the Internet’s stability and usability.

100 C. Kahraman et al. B. Copyright Lawsuits Some people use the Internet to trade copyrighted works, particularly music, vid eo, and software. Copyright owners object to this practice and attempt to discou rage the practice through highly public legal action against participants. C. In adequate Government IT Security Poor safeguarding of personal information could damage the uptake of government services online. For example, on rare occasions, personal records have been foun d at landfill sites, which have caused concern. If people are concerned about go vernment security and about Internet security, they are doubly unlikely to use e -government services. D. Inadequate Government IT Security Concerns are sometimes raised about the availability of “dangerous” information, suc h as bomb-making recipes, on the Internet, and about disinformation or opinion b eing presented as fact. 4.5 SWOT-AHP Analysis Evaluations are made by four experts in a group meeting and they are asked to ma ke a collective group decision both on an evaluation score (Table 2) and on a li nguistic term describing the comparison of SWOT factors and subfactors. 4.5.1 Cr isp SWOT-AHP Analysis When the analysis has been completed, a SWOT matrix can be generated and used as a basis for goal setting, strategy formulation, and implementation. The subfact ors of SWOT analysis are placed in a SWOT matrix as shown in Figure 3.

A SWOT-AHP Application using Fuzzy Concept STRENGTHS S1: Settlement of three laws, namely, e-Signature law, Knowledge Acqui sition law, and Privacy law. S2: e-Transformation projects. S3: Supports from to p level managements of public and private sectors and the NGO representative. WE AKNESSES W1: Lack of access to Internet among certain sections of the population . W2: Lack of finance for capital investment in new technologies. W3: Need to ch ange individual attitudes and organizational cultures. W4: Poor economic power o f citizens and businesses. 101 OPPORTUNITIES O1: A candidate country from the European Union’s information societ y perspective. O2: Efficiency by reducing costs and layers of organizational pro cesses by re-engineering. O3: New and improved services. THREATS T1: Decentraliz ed internet governance. T2: Inadequate government IT security. T3: Copyright law suits. T4: Availability of “Dangerous” Information. Figure 3. SWOT Matrix In the following discussion, the pair-wise comparison matrix among SWOT groups a nd an instance of the comparison matrices of the subfactors are given (Tables 4 and 5). The sample pair-wise comparisons for each level of the hierarchy are giv en in the Appendix. Table 3. Pair-wise Comparison Matrix of the SWOT Groups With respect to the goal Strengths Weaknesses Opportunities Threats Strengths 1 3 7 1/5 Weaknesses 1/3 1 4 1/7 Opportunities 1/7 1/4 1 1/9 Threats 5 7 9 1 Table 4. Pair-wise Comparison Matrix of the Strengths Criteria With respect to strengths group S1 S2 S3 S1 1 7 3 S2 1/7 1 1/5 S3 1/3 5 1

102 C. Kahraman et al. Using the pair-wise comparison matrices given re package, the priorities of the SWOT groups n in Table 6 have been obtained. Table 5. Priorities and Consistency Ratios of Sub Factors SWOT group Priority of SWOT factors the group all ratio factor within priority of the group Strengths 0.109 Weaknesses 0.230 Opportunities 0.623 Threats 0.038 S1. Formation of supervisory and executive committees S2. e-Transformation proje cts 0.06 S3. Support from top level management of the public and private sectors W1. Lack of access to Internet among certain sections of the population W2. Lac k of finance for capital investment in new technologies 0.10 W3. Need to change individual attitudes and organizational cultures, W4. Poor economic power of cit izens and businesses O1. A candidate country for the European Union’s information society perspective, O2. Efficiency by reducing 0.06 costs and layers of organiz ational processes by re-engineering O3. New and improved services. T1. Decentral ized Internet governance, T2. Inadequate government IT 0.05 security T3. Copyrig ht lawsuits T4. Availability of “Dangerous” Information 0.081 0.009 0.731 0.188 0.088 0.077 0.020 0.022 0.636 0.041 0.162 0.010 0.235 0.731 0.060 0.438 0.081 0.049 0.188 0.113 above and the Expert Choice softwa and the subfactors, which are show Comparisons of the Swot Groups and Inconsistency Priority of the Over the factor

0.048 0.658 0.083 0.212 0.002 0.027 0.003 0.009 Figure 4 illustrates the priority weights of the categorized subfactors whose nu merical values are given in Table 6.

A SWOT-AHP Application using Fuzzy Concept 0,45 0,4 0,35 0,3 0,25 0,2 0,15 0,1 0,05 0 103 Rank order S1-S3-S2 W3-W1-W4-W2 O2-O3-O1 T1-T3-T4-T2 Factor weights S W O T SWOT Groups Figure 4. The priority weights of the categorized subfactors with crisp AHP Tabl e 6. The Fuzzy Pair-wise Comparison Matrix of the SWOT Groups GOAL S W O T S (1, 1, 1) (1, 3/2, 2) (3/2, 2, 5/2) (1/2, 2/3, 1) W (1/2, 2/3, 1) (1, 1, 1) (3/2, 2, 5/2) (1/2, 2/3, 1) O (2/5, 1/2, 2/3) (2/5, 1/2, 2/3) (1, 1, 1) (2/5, 1/2, 2/3) T (1, 3/2, 2) (1, 3/2, 2) (3/2, 2, 5/2) (1, 1, 1) Figure 5 illustrates the graphical interpretation of the results of pair-wise co mparisons for SWOT groups and factors. The whole situation is easily observed by referring to Figure 5. The lengths of the lines in the different sectors point out that the weaknesses and opportunities predominate and that currently no spec ific strengths and threats could ruin the new strategy. Figures 6 and 7 show the results of the sensitivity analysis with respect to the goal. From Figure 6, we see the overall weights on the right side of the figure, indicating that S2 (eTransformation projects) is the most important subfactor of all. When we increas e the weight of the strengths group to make it the largest of all the groups, as illustrated on the strengths line, the rank order is S2-T2-W2-S3-O1-T4-W4-S1-T3 -O3-W1-T1-O2-W3. From Figure 7, we see the overall weights on the right side of the figure, indicating that W2 (Lack of finance for capital investment in new te chnologies) is the most important subfactor of all. When we increase the weight of the weaknesses group to make it the largest of all the groups, as illustrated on weaknesses line, the rank order is W2-O1-W4-O3-W1-S2-O2W3-T2-S3-T4-S1-T3-T1.

104 STRENGTHS Weigths C. Kahraman et al. OPPORTUNITIESO1 S2 O2 S1 0.162 0.077 W1 W4 W3 0.027 T3 T4 T1 T2 O3 S3 0.438 Weights W2 WEAKNESSES THREATS Figure 5. Graphical interpretation of the results of pair-wise comparisons of SW OT groups and factors Figure 6. Sensitivity with respect to the goal: SWOT groups

A SWOT-AHP Application using Fuzzy Concept 105 Figure 7. Sensitivity with respect to the goal: SWOT groups 4.5.2 Fuzzy SWOT-AHP Analysis Using Chang’s (1992) extent analysis, we obtained one eigenvector for the SWOT fac tors and four eigenvectors for the subfactors of SWOT. In the following discussi on, the pair-wise comparison matrix among SWOT groups and one sample of the pair -wise comparisons for subfactors of SWOT groups are given (Tables 8 and 9). Some of the pair-wise comparisons are given in the appendix. The eigenvectors of all pair-wise comparisons among SWOT groups and subfactors of SWOT can be seen in T able 7. From Table 7, SS SW 2.90,3.67,4,67 3.40,4.50,5.67 (1 / 22.50,1 18,1 14.20) (1 / 22.50,1 18,1 14.20) (1 / 22.50,1 18,1 14.20) (0.129,0.204,0.329) (0.151,0.250,0.399) (0.244,0.389,0.599) S O 5.50,7.00,8.50 and ST 2.40,2.83,3.67 (1 / 22.50,1 18,1 14.20) (0.107,0.157,0.258) are obtained. Using these vectors, V SW S S 1.00 and other V values are obtained as 1.00, 1.00, 0.74, 0.79, 1.00, 1.00, 0.54, 0.31, 0.53, 1.00, and 0.06, respec tively. Thus, the weight vector from Table 8 is calculated as WG 0.165,0.278,0.5 28,0.030 T .

106 C. Kahraman et al. Table 7. The Fuzzy Pair-wise Comparison Matrix of the Weakness Criteria W W1 W2 W3 W4 W1 (1, 1, 1) (3/2, 2, 5/2) (2/3, 1, 2) (3/2, 2, 5/2) W2 (2/5, 1/2, 2/3) (1, 1, 1) (2/5, 1/2, 2/3) (2/5, 1/2, 2/3) W3 (1/2, 1, 3/2) (3/2, 2, 5/2) (1, 1, 1) (3/2, 2, 5/2) W4 (2/5, 1/2, 2/3) (3/2, 2, 5/2) (2/5, 1/2, 2/3) (1, 1, 1) Table 8. The priority weights of SWOT groups and factors SWOT group Priority of the group SWOT factors Priority of the factor within the group 0.083 Overall priority of the factor 0.0137 Strengths 0.165 Weaknesses 0.278 Opportunities 0.528 S1. Formation of supervisory and executive committees S2. e-Transformation proje cts S3. Support from top-level management of public and private sectors W1. Lack of access to Internet among certain sections of the population W2. Lack of fina nce for capital investment in new technologies W3. Need to change individual att itudes and organizational cultures. W4. Poor economic power of citizens and busi nesses O1. A candidate country from the European Union’s information society persp ective O2. Efficiency by reducing costs and layers of organizational processes b y re-engineering O3. New and improved services, T1. Decentralized Internet gover nance, T2. Inadequate government IT security T3. Copyright lawsuits T4. Availabi lity of “dangerous” information 0.764 0.153 0.052 0.1261 0.0252 0.0145 0.487 0.105 0.1354 0.0292 0.356 0.771 0.0990 0.4071 0.038 0.0201 0.191 0.061 0.565 0.079 0.295

0.1008 0.0018 0.0170 0.0024 0.0089 Threats 0.03

A SWOT-AHP Application using Fuzzy Concept 107 The WW vector 0.05,0.49,0.11,0.36 T . weight from Table 9 is calculated as Table 9. The priority weights of SWOT groups and factors With respect to S1 A1 A2 A3 A4 With respect to S2 A1 A2 A3 A4 With respect to S3 A1 A2 A3 A4 A1 1 5 1/6 1/5 A1 1 1/3 6 1/3 A1 1 3 1/6 1/8 A2 1/5 1 1/8 1/9 A2 3 1 4 1/3 A2 1/3 1 1/5 1/9 A3 6 8 1 1 A3 1/6 1/4 1 1/8 A3 6 5 1 1/2 A4 5 9 1 1 A4 3 3 8 1 A4 8 9 2 1 Inconsistency ratio Priorities of alternatives with respect t o S1 0.352 1.000 0.084 0.083 Priorities of alternatives with respect to S2 0.303 0.188 1.000 0.088 Priorities of alternatives with respect to S3 0.579 1.000 0.1 44 0.080 0.07 Inconsistency ratio 0.10 Inconsistency ratio 0.07 Figure 8 illustrates the priority weights of the categorized subfactors whose nu merical values are given in Table 10. Figure 8. The priority weights of the categorized subfactors

108 C. Kahraman et al. 5. POSSIBLE E-GOVERNMENT STRATEGIES Possible e-government strategies of Turkey are the alternatives for the AHP prob lem above. The e-government initiatives will deliver more and better services to citizens through more effective inter- and intragovernmental teamwork. The gove rnment will deliver these services at a lower cost through the reduction of redu ndant systems and applications. The Office of E-Government and IT will pursue th e following strategies to accomplish the goal: (A1) Simplify work processes to i mprove service to citizens. The individual e-government projects will be driving the migration of systems, data, and processes to a common solution that better meets citizen needs. Agencies will be setting up solutions that cross traditiona l organization “silos,” based on e-business principles of “buy once, use many” and “collec t once, use many.” (A2) Use the annual budget process and other requirements to su pport e-government implementation. There will be a continued consolidation of wo rk plans and investments in technologies acquired by different agencies for like purposes and external-facing transactions platforms. Agencies will continue to reduce redundant spending and improve the return on IT investments through the u se of business cases, capital planning, investment and control process, and thro ugh other means, such as enterprise licensing. (A3) Improve project delivery thr ough development, recruitment, and retention of a qualified IT workforce. The Tu rkish Government will support the efforts to analyze information resource manage ment and personnel needs and assess and upgrade current IT training programs. (A 4) Continue to modernize agency IT management around citizencentered lines of bu siness. The next series of e-government initiatives will drive improvement in th e way the Turkish Government makes and monitors IT investments. The Administrati on has defined an annual cycle for identifying, analyzing, and deploying opportu nities to integrate and consolidate activities along business lines that cross a gency boundaries. The policy of the Administration is that IT transformation wil l be based on consolidation along lines of business and citizen needs: Agencies will have to make the business case for developing a unique solution.

A SWOT-AHP Application using Fuzzy Concept 109 6. EVALUATION OF E-GOVERNMENT STRATEGIES IN TURKEY In this section the importance weights of the e-government strategies are determ ined to find the strategy with the largest weight that should be implemented fir st. 6.1 Crisp AHP Table 9 gives the pair-wise comparison matrices of alternative strategies with r espect to strengths together with the inconsistency ratios. Some other pair-wise comparison matrices of alternative strategies with respect to weaknesses, oppor tunities, and threats are given in the Appendix. Using the Expert Choice softwar e, we obtained the results shown in Figure 9. The rank order of the e-government strategies is A1-A2-A3-A4. In Figures 10 and 11, a sensitivity analysis is give n for SWOT groups and strategy alternatives Total 0,32 0,30 0,28 0,26 0,24 0,22 0,20 0,18 0,16 0,14 0,12 0,10 0,08 0,06 0,04 0,02 0,00 Priorities A1 A2 A3 A4 Alternatives Figure 9. Priorities of e-government strategies

110 C. Kahraman et al. Figure 10. Sensitivity analysis for SWOT groups and strategy alternatives Figure 11. Sensitivity analysis for SWOT groups and strategy alternatives From Figure 10, we see the overall weights on the right side of the figure, whic h indicate that A2 (Use the annual budget process and other requirements to supp ort e-government implementation) is the most

A SWOT-AHP Application using Fuzzy Concept 111 important strategy of all. When we increase the weight of the threats group to m ake it the largest of all the groups, as illustrated on the threats line, the ra nk order is A2-A3-A1-A4. From Figure 11, we see the overall weights on the right side of the figure, which indicate that A3 (Improve project delivery through de velopment, recruitment and retention of a qualified IT workforce) is the most im portant strategy of all. When we increase the weight of the strengths group to m ake it the largest of all the groups, as illustrated on the strengths line, the rank order is A3-A2-A1-A4. 6.2 Fuzzy AHP A sample pair-wise comparison matrix of alternative strategies with respect to s ubfactors is given in Tables 10 and 11. Based on Chang’s (1992) extent analysis, 1 4 eigenvectors for the e-government strategies with respect to the subfactors ar e obtained and given in Table 11 and the overall result is given in Table 12. Table 10. The Fuzzy Evaluation of Alternatives with Respect to the Sub-Attribute O1 O1 A1 A2 A3 A4 A1 (1, 1, 1) (1/2, 2/3, 1) (2/5, 1/2, 2/3) (1/3, 2/5, 1/2) A2 (1, 3/2, 2) (1, 1, 1) (1/2, 2/3, 1) (1/2, 2/3, 1) A3 (3/2, 2, 5/2) (1, 3/2, 2) (1, 1, 1) (1/2, 2/3, 1) A4 (2, 5/2, 3) (1, 3/2, 2) (1, 3/2, 2) (1, 1, 1) Table 11. Weights of Attributes and Scores of Alternatives—Summary S 0.165 S1 A1 A2 A3 A4 W 0.278 S2 0.16 0.17 0.54 0.12 S3 0.24 0.71 0.05 0.00 W1 0.42 0.13 0.09 0.36 W2 0.25 0.68 0.00 0.07 W3 0.00 0.05 0.39 0.56 W4 0.00 0.61 0.00 0.39 O 0.528 O1 0.52 0.30 0.17 0.01 O2 0.56 0.05 0.00 0.39 O3 0.00 0.00 0.61 0.39 T 0.030 T1 0.56 0.34 0.04 0.06 T2 0.00 0.64 0.36 0.00 T3 0.03 0.15 0.71 0.11 T4 0.15 0.00 0.52 0.32 0.083 0.764 0.153 0.052 0.487 0.105 0.356 0.771 0.038 0.191 0.061 0.565 0.079 0. 295 0.32 0.68 0.00 0.00 The W AO1 weight vector T from Table 12 is calculated

as 0.52,0.30,0.17,0.01 .

112 Table 12. Results of Fuzzy AHP C. Kahraman et al. Priority Weights A2 A1 A3 A4 0.341 0.298 0.223 0.139 7. CONCLUSION Electronic government is no longer just an option but a necessity for countries aiming for better governance. People and policies play the primary role in makin g e-government a success. The framework explained in this chapter provides a dir ection for consideration of the evaluation of egovernment strategies. The case s tudy of Turkey provides an illustrative reference for the strategy evaluation. T his model would be beneficial for evaluating any other e-government strategies i n the country and for comparing its priority with the other e-government strateg ies. The selection of various SWOT factors depends on the system profile, the ty pe of services being offered, and the profile of the citizen being served. The q ualitative analysis of these factors and strategies is highly subjective and may differ from one expert to another. Two different SWOT-AHP approaches (the crisp and the fuzzy cases) to the e-government strategy selection produced different rankings but close priority weights. That was caused by the type of information gathered from the experts. In the first method, they are asked to agree on preci se values about alternative strategies, SWOT factors, and subfactors, whereas in the second what they needed to do was to agree on linguistic terms, which expre ss their perceptions on alternative strategies, SWOT factors, and subfactors. Th e strategies “simplify work processes to improve services to citizens” and “use the an nual budget process and other requirements to support E-Government implementatio n” have been found to be the two most important strategies for e-government in Tur key by both methods. New strategies may be proposed and added to the SWOTAHP ana lysis. For additional research, the combination of SWOT and AHP may be changed t o compare the results of this work with the ones of SWOT-TOPSIS, SWOT-Scoring, o r SWOT-ELECTRE.

A SWOT-AHP Application using Fuzzy Concept 113 REFERENCES Akman, I., Yazici, A., and Arifo lu, A., 2002, E-government: Turkey Profile, Pro ceedings of International European Conference on e-government, Oxford, U.K, pp. 27–39. Akman, I., Yazici, A., Mishra, A., Arifo lu, A., 2005, E-Government: A glob al view and an empirical evaluation of some attributes of citizens, Government I nformation Quarterly, 22: 239–257. Chang, D-Y., 1996, Applications of the extent a nalysis method on fuzzy AHP, European Journal of Operational Research, 95: 649 6 55. Chang, D-Y., 1992, Extent analysis and synthetic decision, Optimization Tech niques and Applications, 1, World Scientific, Singapore, 352. Chen, Y.C., and Ga nt, J., 2001, Transforming local e-government services: the use of application s ervice providers, Government Information Quarterly, 18: 343 355. Gil-Garcia, J.R ., and Pardo, T.A., 2005, E-government success factors: Mapping practical tools to theoretical foundations, Government Information Quarterly, 22: 187 216. Gupta , M.P., and Jana, D., 2003, E-government evaluation: a framework and case study, Government Information Quarterly, 30: 365 387. Kahraman, C., Cebeci, U., and Ru an, D., 2004, Multi-attribute comparison of catering service companies using fuz zy AHP:The case of TURKEY, International Journal of Production Economics, 87: 17 1–184. Kaylor, C., Deshazo, R., and Van Eck, D., 2001, Gauging e-government: a rep ort on implementing services among American cities, Government Information Quart erly, 18: 293 307. Kurttila, M., Pesonen, M., Kangas, J., and Kajanus, M., 2000, Utilzing the analytic hierarchy process (AHP) in SWOT analysis—a hybrid method an d its application to a forecast-certification case, Forecast Policy and Economic s, 1: 41 52. Layne, K., and Lee, J., 2001, Developing fully functional E-governm ent: a four stage model, Government Information Quarterly, 18: 122 136. OECD, 20 03, The e-government imperative: Main findings. Vaidvay, O.S., and Kumar, S., 20 06, Analytic hierarchy process: An overview of applications, European Journal of Operational Research, 169: 1–29. Reddick, C.G., 2004, A two-stage model of e-gove rnment growth: Theories and empirical evidence for U.S. cities, Government Infor mation Quarterly, 21: 51 64. Saaty, T.L., 1977, A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology, 15(3): 234 281. Saaty, T.L., 1980, The Analytic Hierarchy Process, McGraw-Hill, New York. Saaty, T.L., 1993, The analytic hierarchy process: a 1993 overview, Central European J ournal of Operation Research and Economics, 2(2): 119 137. Saaty, T.L., and Varg as, L.G., 2001, Models, Methods, Concepts and Applications of the Analytic Hiera rchy Process, Kluwer Academic Publishers, Boston, MA. United Nations / and The A merican Society for Public Administration (MN/ASPA), 2002, Benchmarking e-govern ment: A global perspective, UN/ASPA: New York. Weihrich, H., 1982, The TOWS matr ix–a tool for situation analysis, Long Range Planning, 15(2): 54 66. Zadeh, L., 19 65, Fuzzy sets, Information Control, 8: 338 353. Zimmermann, H.-J., 1994, Fuzzy Set Theory and Its Applications, Kluwer Academic Publishers, Boston, MA.

114 C. Kahraman et al. APPENDIX Some sample pair-wise comparison matrices for each level of the hierarchy are gi ven as follows: Table A.1. Crisp Pair-wise Comparison Matrix of the Opportunities With respect to opportunities group O1 O2 O3 O1 1 1/7 1/5 O2 7 1 3 O3 5 1/3 1 Table A.2. Crisp Pair-wise Comparison Matrix of the Threats With respect to threats group T1 T2 T3 T4 T1 1 9 2 6 T2 1/9 1 1/7 1/5 T3 1/2 7 1 3 T4 1/6 5 1/3 1 Table A.3. The Fuzzy Pair-wise Comparison Matrix of the Opportunities With respect to GOAL O1 O2 O3 O1 (1, 1, 1) (1/3, 2/5, 1/2) (2/5, 1/2, 2/3) O2 (2 , 5/2, 3) (1, 1, 1) (2/3, 1, 2) O3 (3/2, 2, 5/2) (1/2, 1, 3/2) (1, 1, 1) Table A.4. The Fuzzy Pair-wise Comparison Matrix of the Threats With respect to GOAL T1 T2 T3 T4 T1 (1, 1, 1) (3/2, 2, 5/2) (2/3, 1, 2) (1, 3/2, 2) T2 (2/5, 1/2, 2/3) (1, 1, 1) (2/5, 1/2, 2/3) (1/3, 2/5, 1/2) T3 (1/2, 1, 3/2 ) (3/2, 2, 5/2) (1, 1, 1) (3/2, 2, 5/2) T4 (1/2, 2/3, 1) (2, 5/2, 3) (2/5, 1/2, 2/3) (1, 1, 1)

A SWOT-AHP Application using Fuzzy Concept 115 Table A.5. The Pair-wise Comparisons of Alternative Strategies with Respect to t he Opportunities With respect to O1 A1 A2 A3 A4 Inconsistency ratio Normalized priorities of alte rnatives with respect to O1 1.000 0.500 0.408 0.204 Normalized priorities of alt ernatives with respect to O2 1.000 0.145 0.118 0.729 Normalized priorities of al ternatives with respect to O3 0.122 0.128 1.000 0.730 A1 A2 A3 A4 With respect to O2 1 1/2 1/3 1/4 A1 2 1 1 1/3 A2 3 1 1 1/2 A3 4 3 2 1 A4 0.02 Inconsistency ratio A1 A2 A3 A4 With respect to O3 1 1/8 1/5 1/2 A1 8 1 1/2 7 A2 5 2 1 7 A3 2 1/7 1/7 1 A4 0.08 Inconsistency ratio A1 A2 A3 A4 1 1 7 7 1 1 6 7 1/7 1/6 1 1/2 1/7 1/7 2 1 0.03 Table A.6. The Pair-wise Comparisons of Alternative Strategies with Respect to T hreats With respect to T1 A1 A2 A3 A4 Inconsistency ratio Normalized priorities of alternatives with respect to T1 1.000 0.647 0.288 0.228 Normalized priorities of alternatives with respect to T2 0.106 1.000 0.611 0.122 A1 A2 A3 A4 With respect to T2 1 1/2 1/3 1/4 A1 2 1 1/2 1/4 A2 3 2 1 1 A3

4 4 1 1 A4 0.03 Inconsistency ratio A1 A2 A3 A4 1 9 7 1 1/9 1 1/2 1/7 1/7 2 1 1/5 1 7 5 1 0.01

116 With respect to T3 A1 A2 A3 A4 Inconsistency ratio C. Kahraman et al. Normalized priorities of alternatives with respect to T3 0.151 0.317 1.000 0.214 Normalized priorities of alternatives with respect to T4 0.222 0.108 1.000 0.44 5 A1 A2 A3 A4 With respect to T4 1 2 5 2 A1 1/2 1 4 1/2 A2 1/5 1/4 1 1/5 A3 1/2 2 5 1 A4 0.03 Inconsistency ratio A1 A2 A3 A4 1 1/3 6 2 3 1 5 6 1/6 1/5 1 1/3 1/2 1/6 3 1 0.08 Table A.7. The Fuzzy Pair-wise Comparisons of Alternatives with Respect to Oppor tunities With respect to O2 A1 A2 A3 A4 With respect to O3 A1 A2 A3 A4 A1 (1, 1, 1) (1/3, 2/5, 1/2) (1/3, 2/5, 1/2) (1/2, 2/3, 1) A1 (1, 1, 1) (1, 1, 1) (2, 5/2, 3) (3/2 , 2, 5/2) A2 (2, 5/2, 3) (1, 1, 1) (1/2, 1, 3/2) (3/2, 2, 5/2) A2 (1, 1, 1) (1, 1, 1) (2, 5/2, 3) (3/2, 2, 5/2) A3 (2, 5/2, 3) (2/3, 1, 2) (1, 1, 1) (3/2, 2, 5/ 2) A3 (1/2, 2/3, 1) (1/2, 2/3, 1) (1, 1, 1) (1/2, 2/3, 1) A4 (1, 3/2, 2) (2/5, 1 /2, 2/3) (2/5, 1/2, 2/3) (1, 1, 1) A4 (2/5, 1/2, 2/3) (2/5, 1/2, 2/3) (1, 3/2, 2 ) (1, 1, 1) Table A.8. The Fuzzy Pair-wise Comparisons of Alternatives with Respect to Threa ts With respect to T1 A1 A2 A3 A4 With respect to T2 A1 A2 A3 A4 A1 (1, 1, 1) (1/2, 2/3, 1) (1/3, 2/5, 1/2) (1/3, 2/5, 1/2) A1 (1, 1, 1) (5/2, 3, 7/2) (2, 5/2, 3) (1/2, 1, 3/2) A2 (1, 3/2, 2) (1, 1, 1) (1/2, 2/3, 1) (2/5, 1/2, 2/3) A2 (2/7, 1/ 3, 2/5) (1, 1, 1) (2/5, 1/2, 2/3) (2/7, 1/3, 2/5) A3 (2, 5/2, 3) (1, 3/2, 2) (1, 1, 1) (2/3, 1, 2) A3 (1/3, 2/5, 1/2) (3/2, 2, 5/2) (1, 1, 1) (1/3, 2/5, 1/2) A4 (2, 5/2, 3) (3/2, 2, 5/2) (1/2, 1, 3/2) (1, 1, 1) A4 (2/3, 1, 2) (5/2, 3, 7/2) (2, 5/2, 3) (1, 1, 1)

A SWOT-AHP Application using Fuzzy Concept With respect to T3 A1 A2 A3 A4 With respect to T4 A1 A2 A3 A4 A1 (1, 1, 1) (1, 3 /2, 2) (5/2, 3, 7/2) (1/2, 1, 3/2) A1 (1, 1, 1) (1/2, 2/3, 1) (3/2, 2, 5/2) (1, 3/2, 2) A2 (1/2, 2/3, 1) (1, 1, 1) (3/2, 2, 5/2) (2/3, 1, 2) A2 (1, 3/2, 2) (1, 1, 1) (3/2, 2, 5/2) (3/2, 2, 5/2) A3 (2/7, 1/3, 2/5) (2/5, 1/2, 2/3) (1, 1, 1) ( 1/3, 2/5, 1/2) A3 (2/5, 1/2, 2/3) (2/5, 1/2, 2/3) (1, 1, 1) (2/5, 1/2, 2/3) A4 ( 2/3, 1, 2) 117 (1/2, 1, 3/2) (2, 5/2, 3) (1, 1, 1) A4 (1/2, 2/3, 1) (2/5, 1/2, 2/3) (3/2, 2, 5/ 2) (1, 1, 1)

FUZZY OUTRANKING METHODS: RECENT DEVELOPMENTS Ahmed Bufardi, Razvan Gheorghe, and Paul Xirouchakis Institute of Production and Robotics, Ecole Polytechnique Fédérale de Lausanne (EPFL ), Switzerland Abstract: The main objective of this chapter is to account for the most recent development s related to fuzzy outranking methods with a particular focus on the fuzzy outra nking method developed by the authors. The valued outranking methods PROMETHEE a nd ELECTRE III, which are the outranking methods the most used for application i n real-life multi-criteria decision aid problems, are also presented. The descri ption of the general outranking approach is provided. Outranking method, fuzzy o utranking relation, pair-wise comparison, multicriteria decision aid Key words: 1. INTRODUCTION Outranking methods form one of the main families of methods in multicriteria dec ision aid (MCDA). Other important methods are multi-attribute utility theory (MA UT) methods, interactive methods, and the analytic hierarchy process (AHP). It i s worth recalling that the first outranking method called ELECTRE I was develope d by Bernard Roy and published in 1968. Since then, a series of outranking metho ds were developed mainly during the 1970s and 1980s. Among them we can quote ELE CTRE II (Roy and Bertier, 1973), ELECTRE III (Roy, 1978), QUALIFLEX (Paelinck, 1 978), ORESTE (Roubens, 1982; Pastijn and Leysen, 1989), ELECTRE IV (Roy and Hugo nnard, 1982), MELCHIOR (Leclercq, 1984), PROMETHEE I and II C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 119

120 A. Bufardi et al. (Brans and Vincke, 1985), TACTIC (Vansnick, 1986), MAPPACC (Matarazzo, 1986), an d PRAGMA (Matarazzo, 1986). The outranking methods are based on the construction and the exploitation of an outranking relation. The underlying idea consists of accepting a result less rich than the one yielded by multi-attribute utility th eory by avoiding the introduction of mathematical hypotheses that are too strong and asking the decision maker some questions that are too intricate (Vincke, 19 92a). The concept of an outranking relation is introduced by Bernard Roy who is the founder of outranking methods. According to Roy (1974), an outranking relati on is a binary relation S defined on the set of alternatives A such that for any pair of alternatives (a,b) A A: aSb if, given what is known about the preferenc es of the decision maker, the quality of the evaluations of the alternatives and the nature of the problem under consideration, there are sufficient arguments t o state that the alternative a is at least as good as the alternative b, while a t the same time no strong reason exists to refuse this statement. In contrast to the other methods, the outranking methods have the characteristic of allowing i ncomparability between alternatives. This characteristic is important in situati ons where some alternatives cannot be compared for one or another reason. Accord ing to Siskos (1982), incomparability between two alternatives can occur because of a lack of information, inability of the decision maker to compare the two al ternatives, or his refusal to compare them (Siskos, 1982). In contrast to the va lued outranking methods that are well documented in the literature and have been intensively used in practice since 1978 with the publication of ELECTRE III, th e fuzzy outranking methods are very recent and are not well documented in the li terature, and this is one of the motivations for the redaction of this chapter. The chapter is structured as follows. The main elements of a general outranking approach are described in Section 2. Section 3 is devoted to the presentation of the PROMETHEE and ELECTRE III, which are the main valued outranking methods con sidered in both theory and applications. The fuzzy outranking methods are presen ted in Section 4. Some concluding remarks are given in Section 5. 2. THE OUTRANKING APPROACH An outranking method is applicable for MCDA problems where the elements of a fin ite set of alternatives A = {a1, a2, …, an} have to be

Fuzzy Outranking Methods 121 compared on the basis of the preferences of the decision maker regarding their p erformances with respect to the elements of a finite set of criteria F = {g1, g2 , …, gm}. It is assumed that each alternative ai, i = 1, …, n can be evaluated with respect to each criterion gj, j = 1, …, m. The evaluations can be quantitative or qualitative. They can also be deterministic or nondeterministic. In the nondeter ministic case, they can be fuzzy or stochastic. The objective of outranking meth ods is provide decision aid to decision makers in the form of a subset of “best” alt ernatives or a partial or complete ranking of alternatives (Pasche, 1991). Accor ding to Roy (1991), the preferences in the outranking concept are determined at two different levels as follows: Level of preferences restricted to each criteri on. For example, to each criterion gj, it is possible to associate a restricted outranking relation Sj such that for any two alternatives a and b in A: aS j b a, with respect to g j , is at least as good as b (1) Level of comprehensive preferences where all criteria are taken into account. Th e meaning of an outranking relation is given in Section 1. However, there is a n eed for a set of conditions to recognize whether a given binary relation can be an outranking relation. The following definition is provided in (Perny and Roy, 1992) DEFINITION 1. A fuzzy relation Sj defined on A2 is said to be a monocriter ion outranking index for a criterion gj if a real-valued function tj, exists def ined on A 2 , verifying Sj(a, b) = tj(aj, bj) for all a and b in A with aj and b j being the crisp scores of a and b on criterion gj such that: y0 x0 z , t j x , y0 is a nondecreasing function of x, , t j x0 , y is a nonincreasing f unction of y, ,t j z, z 1. It is worth noticing that the three conditions in this definition are also valid for fuzzy outranking relations constructed from fuzzy evaluations on criteria a nd for global outranking relations.

122 A. Bufardi et al. In the literature, confusion abounds regarding valued and fuzzy outranking relat ions, and they are often used interchangeably. Even if the valued and fuzzy outr anking relations are similar from a mathematical point of view, they represent t wo different situations: The valued outranking relation represents a crisp situa tion, and the value S (a, b) 0, 1 represents the intensity with which the altern ative a outranks alternative b and S (a, b) is constructed from crisp evaluation s of alternatives a and b. The valued outranking relation represents a fuzzy sit uation, and the value S(a,b) 0, 1 represents the degree with which the alternati ve a is R-related to b and S(a,b) is constructed from fuzzy evaluations of alter natives a and b. An outranking method is composed of two main phases that are th e construction of a global outranking relation and the exploitation of this rela tion. The construction phase is composed of two main steps: Construction of an o utranking relation or related relations such as concordance and discordance indi ces with respect to each criterion, The aggregation of the single outranking rel ations into a global outranking relation. The exploitation phase of a valued/fuz zy outranking method can be dealt with in three different ways (Fodor and Rouben s, 1994): Transformation of the valued/fuzzy outranking relation into another va lued/fuzzy relation having particular properties such as transitivity that are i nteresting for the ranking of alternatives, Determination of a crisp relation cl osed to the valued/fuzzy outranking relation and having specific properties, Use of a ranking procedure to obtain a score function as it is the case for PROMETH EE and ELECTRE III methods. A detailed study of the exploitation phase in the ca se of crisp relations is provided in (Vincke, 1992b).

Fuzzy Outranking Methods 123 3. VALUED OUTRANKING METHODS The outranking methods that are the most used for application in real-life MCDA problems are ELECTRE III and PROMETHEE, which are valued outranking methods sinc e they are based on the construction and exploitation of a valued “outranking rela tion.” ELECTRE stands for “ELimination Et Choix Traduisant la REalité,” and PROMETHEE st ands for “Preference Ranking Organization METHod for Enrichment Evaluations.” 3.1 ELECTRE III ELECTRE III is an outranking method proposed by Roy (1978) to deal with multi-cr iteria decision-making situations in which a finite set of alternatives should b e ranked from the best to the worst. It is composed of the following steps: The construction of a valued outranking relation; The construction of two complete p reorders based on descending and ascending distillation chains; The comparison o f the two complete preorders in order to elaborate a final ranking of the altern atives. This comparison leads to a partial preorder in which it is possible that some alternatives are incomparable. 3.1.1 The Construction Phase of ELECTRE III Let A = {a1, a2, …, an} be a finite set of n alternatives and F = {g1, g2, …, gm} a set of m criteria on which the alternatives in A will be evaluated. Without loss of generality, the criteria can be assumed to be maximizing, i.e., the higher t he performance of an alternative on a criterion is, the better the alternative i s. ELECTRE III is based on the definition of a valued outranking relation S such that for each ordered pair of alternatives (a,b), S(a,b) [0, 1] represents the degree to which alternative a is at least as good as alternative b (the degree t o which alternative a is not worse than alternative b). 3.1.1.1 Single Criterion Relations With each criterion gj (j = 1, …, m) are associated four parameters: a weight wj, a preference threshold pj, an indifference threshold qj, and a veto

124 A. Bufardi et al. threshold vj. It is naturally assumed that for each alternative a: qj(gj(a)) pj( gj(a)) vj(gj(a)). With each criterion gj (j = 1, …, m) are associated a concordanc e index cj and a discordance index dj as follows which are shown in Figures 1 an d 2 respectively. 1 if g j (a ) q j (g j (a )) c j (a, b) 0 if g j (a ) p j (g j (a )) p j (g j (a )) g j (b), g j (b), otherwise (2) g j (a ) g j (b) p j (g j (a )) q j (g j (a )) * may occur only in the case when qj(gj(a)) pj(gj(a)). 0 if g j (b) d j ( a, b) 1 if g j (b) v j ( g j (a )) g j (a) p j ( g j ( a )), g j (a ) v j ( g j (a )), p j ( g j (a )) p j ( g j (a )) otherwise* (3) g j (b) g j (a ) * may occur only in the case when pj(gj(a)) cj (a,b) vj(gj(a)). 1 0 gj (a) gj (a)+qj (gj(a)) gj (a)+pj (gj(a)) Figure 1. Concordance index of gj

Fuzzy Outranking Methods dj (a,b) 125 1 0 gj (a) gj (a)+pj (gi(a)) gj (a)+vj (gi(a)) gj (b) Figure 2. Discordance index of gj 3.1.1.2 Global Valued Outranking Relation For each ordered pair of alternatives (a,b), a concordance index c(a,b) is computed in the following way: c ( a, b) 1 W m m w j c j (a, b), where W j 1 j 1 wj (4) It is worth noticing that c(a,b) = 1 means that there is no criterion for which alternative b is better than alternative a and c(a,b) = 0 means that alternative a is worse than alternative b for all criteria. The valued outranking relation S is constructed from the concordance and discordance indices. For each ordered pair of alternatives (a,b) A A, S(a,b) is defined in the following way: c(a, b) if d j (a, b) c(a, b), S ( a, b) c ( a, b) 1 d j ( a, b) j J ( a ,b ) j 1, ..., m otherwise (5) 1 c ( a, b) where J = {j {1, …, m}/dj(a, b) > c(a, b)}. The degree of outranking is equal to t he concordance index when no criterion is discordant. When at least one criterio n is discordant, the degree of outranking is equal to the concordance index mult iplied by a factor lowering the concordance index in function of the importance of the discordances. At the extreme, when dj(a,b) = 1 for some criterion gj, S(a ,b) = 0. Thus, for each ordered pair of alternatives (a,b) A A, 0 S(a,b) 1. S is a valued outranking relation.

126 A. Bufardi et al. 3.1.1.3 The Exploitation Phase of ELECTRE III The second step in ELECTRE III con sists in defining two complete preorders from the descending and the ascending d istillation chains. Let 0 max S (a, b) . At each iteration of the descending or ascending a,b A distillation chain, a discrimination threshold s( ) and a crisp relation D are defined such that: D ( a, b) 1 if S (a, b) 0 otherwise S( ) (6) For each alternative a, a qualification score Q(a) is computed as the number of alternatives that are outranked by a (number of alternatives b such that D(a,b) = 1) minus the number of alternatives, which outrank a (number of alternatives b such that D(b,a) = 1). ELECTRE III provides the decision makers with two comple te preorders. The first preorder is obtained in a descending manner starting wit h the selection of the alternatives with the best qualification score and finish ing with the selection of the alternatives having the worst qualification score. The second preorder is obtained in an ascending manner, first selecting the alt ernatives with the worst qualification score and finishing with the assignment o f the alternatives that have the worst qualification score. 3.1.1.4 Descending D istillation Chain In the descending procedure, the set of alternatives having th e largest qualification score constitutes the first distillate and is denoted as D1. If D1 contains only one alternative, the previous procedure is performed in the set A\D1. Otherwise it is applied to D1 and a distillate D2 will be obtaine d. If D2 is a singleton, then the procedure is applied in D1\D2 if it is not emp ty; otherwise the procedure is applied in D2. This procedure is repeated until t he distillate D1 is completely explored. Then, the procedure starts exploring A\ D1 in order to find a new distillate. The procedure is repeated until a complete preorder of the alternatives is obtained. This procedure is called the descendi ng distillation chain because it starts with the alternatives having the highest qualification and ends with the alternatives having the lowest qualification. T he result of the descending procedure is a set of classes C 1 , C 2 , … , C k with k n. The alternatives belonging to the same class are considered to be ex-æquo (i ndifferent), and an alternative belonging to a class outranks all the alternativ es belonging to classes with higher indices. Thus, a first complete preorder of the alternatives is obtained.

Fuzzy Outranking Methods 127 3.1.1.5 Ascending Distillation Chain The ascending procedure is the same as the descending procedure except that the criterion of selecting the alternatives is based on the principle of the lowest qualification. The result of this procedure is a set of classes C 1 , C 2 , …, C h with h n. These classes are written in suc h a way that two alternatives in the same class are considered to be ex-æquo and a n alternative belonging to a class outranks all the alternatives belonging to cl asses with lower indices. Thus, a second complete preorder of the alternatives i s obtained. 3.1.1.6 Partial Preorder of ELECTRE III The result of ELECTRE III is a partial preorder of the alternatives based on the comparison of the two compl ete preorders obtained by means of the descending and the ascending distillation chains. 3.1.2 Main Features of ELECTRE III ELECTRE III has many interesting features among which we can quote: Handling imp recise and uncertain information about the evaluation of alternatives on criteri a by using indifference and preference thresholds, Consideration of incomparabil ity between alternatives; when two alternatives cannot be compared in terms of p reference or indifference, they are considered to be incomparable. Indeed, somet imes the information available is insufficient to decide whether two alternative s are indifferent or one is preferred to the other, Use of veto thresholds. This is very important for some problems such as those involving environmental and s ocial impacts assessment. According to Rogers and Bruen (1998), within an enviro nmental assessment, it seems appropriate to define a veto as the point at which human reaction to the criterion difference becomes so adverse that it places an “e nvironmental stop” on the option in question. The same can be said about social im pact assessment. ELECTRE III is widely used for different real-world application s such as environmental impact assessment and selection problems in various doma ins. Examples of these applications can be found in Augusto et al. (2005), Becca li et al. (1998), Bufardi et al. (2004), Cote and Waaub (2000), Hokkanen and Sal minen (1994, 1997), Kangas et al. (2001), Karagiannidis and Moussiopoulos (1997) , Maystre et al. (1994), Rogers and Bruen (2000),

128 A. Bufardi et al. Roy et al. (1986), Teng and Tzeng (1994), and Tzeng and Tsaur (1997). The list i s not exhaustive and is given just for illustrative purposes to show the varied and numerous applications of the ELECTRE III method. 3.1.3 Illustrative Example This illustrative example is taken from Bufardi et al. (2004). The problem consi dered consists of selecting the best compromise end-of-life (EOL) alternative to treat a vacuum cleaner at its EOL. Theoretically the number of potential EOL al ternatives that can be considered is very high. In general only a few EOL altern atives are interesting. Users have their own ways for defining EOL alternatives depending on activity, objectives, experience and constraints from market, legis lation, and available technology. In this illustrative example, five EOL alterna tives are considered and described as follows. EOL alternative 1 consists of rec ycling as much as possible and incinerating the rest. EOL alternative 2 consists of recycling only parts with benefits and incinerating the rest. EOL alternativ e 3 consists of recycling all metals that cannot be incinerated and incinerating all the rest. EOL alternative 4 consists of reusing the motor, recycling metals , and incinerating the rest. EOL alternative 5 consists of landfilling all. The five EOL alternatives are presented in Table 1. The criteria used for the evalua tion of EOL alternatives are presented in Table 2. The detailed description of t he environmental criteria presented in Table 2 can be found in Goedkoop and Spri ensma (2000). Once the EOL alternatives EOL alternative 4 EOL alternative 4 EOL alternative 2 EOL alternative 2 EOL alternative 3 EOL alternative 1 EOL alte rnative 3 EOL alternative 1 EOL alternative 5 (a) EOL alternative 5 (b) Figure 3. Partial and median preorder

Fuzzy Outranking Methods 129 and criteria are selected, each EOL alternative is evaluated with respect to eac h criterion as shown in Table 3. The results of applying ELECTRE III can be pres ented in the form of a partial preorder as shown in Figure 3a or a median preord er as shown in Figure 3(b). Table 1. The EOL Alternatives EOL alternatives 1 2 3 4 5 1 Dust bin REC INC INC INC LND 2 2 x Inner Cover REC INC INC INC LND 3 Inner filter asb INC INC INC INC LND 4 Dust bin cover INC INC INC INC LND 5 Lock ring REC INC INC INC LND 6 Spri ng REC INC REC REC LND 7 Power button cover (+ button) REC INC INC INC LND 8 Spr ing REC INC REC REC LND 9 Upper VC case REC INC INC INC LND 10 Suction tube REC INC INC INC LND 11 Suction tube sealing INC INC INC INC LND 12 Intermediate tube REC INC INC INC LND 13 Cables REC REC INC INC LND 14 Valve INC INC INC INC LND 15 Intern sealing 1 INC INC INC INC LND 16 Intern sealing 2 INC INC INC INC LND 17 Spring REC INC REC REC LND 18 Middle REC INC INC INC LND 19 Hepa cover REC IN C INC INC LND INC INC INC INC LND 20 Hepa filter 21 Cable coil cover REC INC INC INC LND 22 Cable coil INC INC INC INC LND 23 Cable REC REC INC INC LND 24 Motor Lock ring REC INC INC INC LND 25 Motor bottom seal INC INC INC INC LND 26 Motor sealing INC INC INC INC LND 27 Motor foam INC INC INC INC LND REC REC REC REM L ND 28 Motor 29 Motor housing half 2 REC INC INC INC LND 30 Motor housing Filter INC INC INC INC LND 31 Motor housing half 1 REC INC INC INC LND 32 32 Motor hous ing seal INC INC INC INC LND 33 Wheels INC INC INC INC LND 34 Lower VC case REC INC INC INC LND 35 Spring REC INC REC REC LND * remanufacturing/reuse (REM), rec ycling (REC), incineration with energy recovery (INC), disposal to landfill (LND ) No. Component/subassembly

130 Table 2. List of Criteria Category Economic Environmental Criterion EOL Treatmen t Cost (C) Human Health (HH) Ecosystem Quality (EQ) Resources (R) Unit [CHF] [DA LY] [PDF*m2yr] [MJ surplus] A. Bufardi et al. Direction of preferences Minimization Minimization Minimization Minimization Table 3. Evaluation of EOL Alternatives Human health (HH) [DALY] 1.08E-05 0.951E -05 0.724E-05 2.90E-05 0.0271E-05 Ecosystem quality (EQ) [PDF*m2yr] 0.471 0.962 0.896 2.02 0.0103 Resources (R) [MJ surplus] 18.1 7.49 6.76 36.8 0.0101 EOL trea tment cost (C) [CHF] 0.644125 0.10601 0.01108 4.86022 0.38101 EOL alternative 1 EOL alternative 2 EOL alternative 3 EOL alternative 4 EOL alte rnative 5 3.2 PROMETHEE PROMETHEE is a MCDA method based on the construction and the exploitation of a v alued outranking relation (Brans and Vincke, 1985). Two complete preorders can b e obtained by ranking the alternatives according to their incoming flow and thei r outgoing flow. The intersection of these two preorders yields the partial preo rder of PROMETHEE I where incomparabilities are allowed. The ranking of the alte rnatives according to their net flow yields the complete preorder of PROMETHEE I I. 3.2.1 The Construction Phase of PROMETHEE Let A = {a1, a2, …, an} be a finite set of alternatives and F = {g1, g2, …, gm} a fi nite set of criteria on which the alternatives will be evaluated. With each crit erion gj, j = 1, 2, …, m, is assigned a weight pj reflecting its relative importan ce. For each pair of alternatives (a,b) A A, an outranking degree (a,b) is compu ted in the following way: 1 m ( a, b) p j H j ( a, b) (7) P j1 m P j 1 p j and Hj(a,b) are numbers between 0 and 1 that are a function of gj(a) – gj(b). For the computation of Hj(a,b)’s, the decision maker is

Fuzzy Outranking Methods 131 given six forms of curves described in Table 1. It is worth noticing that in Tab le 4, the six functions are described for a maximizing criterion where H(x) = P( a,b) if x 0 and H(x) = P(b,a) if x 0. Table 4. List of Generalized Criteria Type of criterion Analytical definition Sh ape 1 H(x) 1. Usual H ( x) 0 1 if x if x 0 0 0 H(x) x 2. Quasi H ( x) 0 1 if x q otherwise -q 1 x 0 q H(x) 3. Linear preference H ( x) x/p 1 if x p otherwise -p 1 x 0 p 0 4. Level x q x x q p q q p 1 0.5

H(x) H ( x) 0.5 1 x -(p+q) -q 0 q p+q 5. Linear preference H ( x) and indifference area 0 ( x - q) / p 1 q x x q q p 1 H(x) otherwise x -(q+p) -q 0 q q+p H(x) 6. Gaussian H ( x) 0 1- e - x2 / 2 2 x x 0 0 1 x 0

132 A. Bufardi et al. 3.2.2 The Exploitation Phase of PROMETHEE With each alternative are associated two values +(a) and (a). + (a), which is ca lled the outgoing flow and is computed in the following way: (a) b A ( a, b) (8) (a), which is called the incoming flow and is computed in the following way: (a) b A (b , a ) (9) It is worth noticing that +(a) represents the degree by which alternative a outr anks the other alternatives and that (a) represents the degree by which alternat ive a is outranked by the other alternatives. The higher the outgoing flow and t he lower the incoming flow, the better the alternative. The two flows induce the following complete preorders (ranking of the alternatives with consideration of indifference) on the alternatives, where P and I are the preference relation an d indifference relation, respectively: aP b aI b aP b aI b (a) > (b) (a) = (b) (a) < (b) (a) = (b) where P+, I+ refer to the outgoing flows while P , I refer to the incoming flows . By ranking the alternatives in the decreasing order of the numbers (a) and in the increasing order of the numbers (a) , two complete preorders can be obtained . Their intersection yields the partial order of PROMETHEE I as follows: if aP b and aP b aSb a strictly outranks b or aP b and aI b or aI b and aP b (10)

Fuzzy Outranking Methods 133 aIb (a is indifferent to b) if aI b and aI b aJb (a and b are incomparable) otherwise (11) (12) i.e., aSb, bSa and aIb , where “ ” denotes negation. For each alternative a, a net f low (a) can be obtained by subtracting the incoming flow (a) from the outgoing f low +(a); i.e., (a) = +(a) (a). By ranking the alternatives in the decreasing or der of , one obtains the unique complete preorder of PROMETHEE II. 3.2.3 Main Fe atures of PROMETHEE PROMETHEE has many interesting features among which we can quote: It is easy to understand. The mathematical background behind PROMETHEE is not complicated and is easy to understand by the users. This is important for the transparency of th e results, It is easy to use. For each criterion, the decision maker has to fix the weight of this criterion, and at most two parameters of the function are ass ociated with the criterion in order to derive the single-valued outranking relat ion related to this criterion, Consideration of incomparability between alternat ives through PROMETHEE I; when two alternatives cannot be compared in terms of p reference or indifference, they are considered to be incomparable. Indeed, somet imes the information available is insufficient to decide whether two alternative s are indifferent or one is preferred to the other. PROMETHE is an outranking me thod easy to understand and to use. That is why it is widely used for practical MCDA problems in various domains; see, e.g., Al-Rashdan et al. (1999), Anagnosto poulos et al. (2003), Babic and Plazibat (1998), Elevli and Demirci (2004), Geld ermann et al. (2000), Gilliams et al. (2005), Goumas and Lygerou (2000), Hababou and Martel (1998), Kalogeras et al. (2005), Le Téno and Mareschal (1998), Mavrota s et al. (2006), and Petras (1997). The list is not exhaustive and is given just for illustrative purposes.

134 A. Bufardi et al. 4. FUZZY OUTRANKING METHODS In these methods, it is assumed that the evaluations of alternatives on criteria are fuzzy. 4.1 Fuzzy Outranking Method of Gheorghe et al. The fuzzy outranking method presented in this subsection is published in Gheorgh e et al. (2004, 2005). Full details can be found in Gheorghe (2005). 4.1.1 Const ruction of Monocriterion Fuzzy Outranking Relation The construction of the monocriterion fuzzy outranking relation starts by analyz ing the intervals, in our case, the -cuts of fuzzy performance of two alternativ es a and b. Let us consider two normalized and convex fuzzy numbers A and B, rep resenting the performances of alternatives a and b, respectively (Figure 4). Let A and B be the membership functions of A and B, respectively. Each i i i i i-cu t is defined by the interval ( a1 , a2 ) for A and (b1 , b2 ) for B, respectivel y, where i = 1, …, N, with N denoting the number of -cuts considered. A, B 1 A B a i i b i 0 a1 a1 i b1 b1 i a2 i a2 b2 i b2 Figure 4. Fuzzy performances of alternatives a and b The comparison performances of the alternatives a and b at the i-cut level using the mechanisms shown in Figures 5 and 6 are in accordance with common sense and represent two different view points. When the interval a i is entirely on the l eft of the interval b i , there is no doubt that a is worse than b and that the degree of trueness of the proposition “a is not worse than b” is 0. When starting to translate a i to the right and the two intervals overlap, this degree of truene ss increases and reaches the

Fuzzy Outranking Methods 135 i maximum value 1 at the moment when the lower limit (left) of a with the lower li mit (left) of b i (Figure 5). b1 i a1 i a2 i a1 i b1 i a2 i b2 i b2 i is equal 1 a2 i b1 i a2 i b1 i (a2 i a1 i ) Figure 5. The first case of the achievement of a degree of trueness of 1 of the proposition “a is not worse than b” A similar judgment can be performed for the case when the maximum degree of true ness is attained at the moment when the upper limit (right) of a i is equal with the upper limit (right) of b i (Figure 6). Thus the reasoning we have done prev iously is suitable for the case when a higher value of performance is preferred to a lower value, in other words, for the case when we want to maximize the perf ormance value with respect to a criterion. Similar reasoning can be followed for the case of a minimizing criterion. b1 i a1 i a2 i b1 i a1 i 1 b2 i b2 i a2 i a2 i b1 i a2 i b2 i Figure 6. The second case of the achievement of a degree of trueness of 1 of the proposition “a is not worse than b”

136 A. Bufardi et al. DEFINITION 2. For each i-cut level, two left i-cut indices are defined for, resp ectively, the case of maximizing and minimizing criteria as the I to 0 ,1 , wher e I is the set of functions sl _i max and, sl _i min from I all real intervals: 0, sl _i max (a i , b i ) a2 i a2 i 1, 0, sl _i min (a i , b i ) b2 i b2 i 1, a1 i , b1 i b1 i , a1 i a2 i a1 i a1 i b2 i b1 i b1 i b1 i b1 i b1 i a1 i a1 i a1 i b2 i a2 i (13) (14) The right i-cut indices can be defined in a similar way as shown in the followin g definition. DEFINITION 3. For each i-cut level, two right i-cut indices are de fined for, respectively, the case of maximizing and minimizing criteria as the i i I to 0 ,1 such that: functions sr _ max and sr _ min from I 0, i sr _ max (a i , b i ) a2 i b1 i , b1 i b1 i a2 i b2 i a1 i , a1 i a1 i b2 i b1 i a2 i b2 i a1 i b2 i a2 i a2 i b2 i a2 i b2 i 1, 0, (15) i sr _ min (a i , b i ) b2 i a2 i 1, (16)

Fuzzy Outranking Methods 137 DEFINITION 4. For each i-cut level, two right i-cut indices are defined for, res pectively the case of maximizing and minimizing criteria as the i i I to 0 ,1 su ch that: functions smax and, smin from I i smax ( a i , b i ) (1 ) sl _i max ( a i , b i ) i sr _ max ( a i , b i ), a, b A (17) i smin ( a i , b i ) (1 i ) sr _ min ( a i , b i ) sl _i min ( a i , b i ), a, b A (18) [0,1] represents the degree of optimism of the decision The parameter maker (Lio u and Wang, 1992). It allows the decision maker to choose which side of the inte rval is more important. When k increases from 0 to 1, the degree of optimism inc reases, whereas the degree of pessimism decreases. This type of strategy will be called the horizontal strategy. In the remaining of this chapter, the notation s or S are used without the index min or max and refer to the maximization case; however, the related statements are also valid for the minimization case, unles s otherwise stated. PROPOSITION 1. The i-cut indices defined in Definition 4 are fuzzy outranking relations. The transition from a fuzzy outranking relation def ined at the -cut level to a single criterion fuzzy outranking relation requires an aggregation procedure. Observing the case of fuzzy numbers A and B presented in Figure 7, it follows that the upper -cut indices favor B, whereas the lower o nes favor A. A compensative approach gives a certain discrimination power while still using the biggest amount of information contained in the fuzzy representat ion of the performances. This idea was exploited in area compensation methods fo r comparing fuzzy numbers by many authors (Chanas, 1987; Fortemps and Roubens, 1 996; Matarazzo and Munda, 2001; Nakamura, 1986). The basic principle is that som e nonintersecting areas (i.e., upper left and/or right external areas and lower left and/or right external areas in Figure 7) compensate each other. If we see t he previously defined -cut indices as relative intersections, then their aggrega tion can be seen as compensation between relative intersections, which is someho w related to the above-mentioned methods. If for linear membership functions the areas considered are relatively simple to be determined, for nonlinear cases, i t becomes more difficult. In our -cut approach besides

138 A. Bufardi et al. the fact that we can use inputs stated as a set of -cut intervals (which avoids possible necessary re-approximations of the original membership function), we pr event the use of integrals for calculating the areas used by an area compensatio n class of methods. A B 1 B A 0 Figure 7. A complex case of comparison of fuzzy numbers The function used to aggregate the -cut indices is the weighted rootpower mean d efined for all x as follows (Smolíková and Wachowiak, 2002): 1 N i ( xi ) Fw ( x) i 1 N i i 1 (19) Using the aggregation function Fw to aggregate i-cut indices, i = 1, …, N, we obta in single criterion fuzzy outranking relation S as follows: 1 N i ( s i (a i , b i )) N i i 1 S ( A, B) i 1 (20) PROPOSITION 2. The single criterion outranking index defined by the relation (20 ) satisfies the following properties: For any convex and normalized fuzzy number B0, S(A, B0) is a nondecreasing function of A;

Fuzzy Outranking Methods 139 For any convex and normalized fuzzy number A0, S(A0, B) is a nonincreasing funct ion of B; For any fuzzy convex and normalized fuzzy number: S(C, C) = 1; hence S is reflexive. Since the definitions of sl i and sr i allow them to take the val ue “0,” at any -cut level, some of the particular cases of the relation (20) are exc luded: Geometrical mean for = 1 due to the possible division by zero; Product me an for 0, because of the risk of penalty of the result, when an -cut level of S is 0. As our intention is to offer the decision maker a flexible decision instru ment, cases like min or max are also excluded. They are dictatorial aggregators, not allowing for compensation between lower and higher values. Two particular c ases are of special interest for the definition of the single criterion fuzzy ou tranking relation: the weighted arithmetic mean (21) and the weighted square ave rage mean (22). N i s i (a i , b i ) N i i 1 S ( A, B ) i 1 (21) N i ( s i (a i , b i )) N i i 1 2 1 2 S ( A, B) i 1 (22) The consideration of weights for -cut indices makes the final relation more flex ible and offers to the decision maker the possibility to decide on the importanc e of the -cut levels during the aggregation. As we have to deal with an enlarged number of weights, equal with the number of -cuts (which is N), we look to auto matically generate the weights. We will search for a method that can give the po ssibility of changing the weighting vector, such that, for different personaliti es of the decision maker, we can build different weighting vectors. For example, in the case where the decision maker wants to rely his decisions on -cuts

140 A. Bufardi et al. with less uncertainty, he might be able to slide the highest weights to the high est -cuts. Alternatively, one might want to give equal importance to all the -cu ts or to assign higher weights to lower -cuts. Here we will consider the case wh en the weights i increase in a linear manner, so the interpolation of these poin ts is a line. As we want to use the information given by all the -cut indices, t his kind of linearity looks convenient, because with two exceptions (the limit f unctions from this family, which will give 0 for the first -cut, respectively fo r the last one), all the weights will be nonzero. The equation of such a line is : i i c (23) where is the slope of the line and c . Through a series of calculations, using t he three particular cases mentioned above and other conditions, the relation (23 ) becomes i ( ) i 1 N 1 2 1 N i N 1 2 i 1 N (24) If we consider i as a continuous parameter, then function transforms into a i, i 1 N 1 2 1 of two variables, which can be N represented as a surface, as shown in Figure 8. Therefore, for the case of a max imizing criterion, we obtain the following the single criterion fuzzy outranking relation: N S max ( A, B ) i 1

i N 1 2 1 N [(1 ) sl _i max ( a i , b i ) i sr _ max ( a i , b i )] . (25) The expression of the single criterion fuzzy outranking relation for the case of a minimizing criterion can be obtained in a similar way.

Fuzzy Outranking Methods 141 Figure 8. The function (i, ) 4.1.2 Aggregation of Single Fuzzy Outranking Relations Here we are interested in aggregating over le criterion fuzzy outranking relations Sk on S. Using an aggregation operator M, the defined for each pair of alternatives (a, S(a,b) M S1 (a,b), ... , Sn (a,b) . (26) Obviously, S must have the properties of a fuzzy outranking relation, and the fo llowing proposition establishes the minimal conditions that an aggregator should fulfill in order to satisfy it. the set of criteria g1, …, gn, the sing into a global fuzzy outranking relati global fuzzy outranking relation S is b) as follows:

142 A. Bufardi et al. PROPOSITION 3. Any aggregator that satisfies the properties of idempotency and m onotonicity with respect to the integrand, used to aggregate single criterion fu zzy outranking relations, leads to a global fuzzy relation that is a fuzzy outra nking relation. The Choquet integral (Grabisch, 1999; Marichal, 1999) is an aggr egator that satisfies these two properties; consequently, the fuzzy relation obt ained by aggregating single criterion fuzzy outranking relations through the use of a Choquet integral is a fuzzy outranking relation. Considering the Choquet i ntegral as the aggregation operator M, S(A, B) becomes n S ( A, B ) k 1 S( k ) ( A, B ) [ {( k ),...,( n )} {( k 1),...,( n )} ] (27) 4.1.3 Exploitation of the Global Fuzzy Outranking Relation The type of exploitation to be undergone by the global fuzzy outranking relation depends among others on the type of application for which this exploitation is to be used. The problem for which this fuzzy outranking method was developed is one in which a large number of decisions has to be taken and for whose solving a n automated decision-making procedure has to be put in place (e.g., the selectio n of the best EOL option for a large number of nodes in a disassembly tree of pr oduct with a complex assembly structure, the ranking of design concepts accordin g to their lifecycle performance, including their EOL, etc.; see Gheorghe and Xi rouchakis (2006) for a detailed description), but its application goes far beyon d this context. It was shown that the formulation (choice of the best alternativ e) of the exploitation problem (Roy, 1977) is the most suitable. Roubens (1989) defined four generalized choice functions C1, C2, C3, and C4 with all of them be ing in the authors’ opinion, intuitively attractive. As it can be seen from their definition, all these choice functions (and in general all possible choice funct ions) refer to the strength of the chosen alternative(s) over the rest of altern atives, so they measure somehow the domination of selected alternative(s) over t he other or the nondomination of other alternatives on the selected one(s). The superscript “+” is used to denote the choice functions selecting the “best” alternative( s).

Fuzzy Outranking Methods 143 The weak domination of alternative a over all the other alternatives is defined as follows: C1 (a ) 1 b A \{ a } T S ( a, b) (28) where T1 is a t-norm and S(a, b) is the degree, between 0 and 1, to which a is a s good as b. To be in accordance with Orlovsky’s (Orlovsky, 1978) reasoning and te rminology, C1 (a) can be interpreted as the degree of weak domination of a over all the other alternatives in A. The choice set is given by C1 ( A, S ) a A C1 ( a ) max C1 (b) b A (29) The weak nondomination of a by all the other alternatives is defined as follows: C2 ( a ) 1 b A \{ a } T [1 S (b, a )] (30) C2 (a) is interpreted as the degree of weak nondomination of a by all the other alternatives in A. The choice set is given by C2 ( A, S ) a A C2 ( a ) max C2 (b) b A (31) The strict domination of a over all the other alternatives is defined as follows : C3 ( a ) 1 b A \{ a} T P ( a, b) (32) C3 (a) represents the degree of strict domination of a over b. P is the strict p reference relation, and it is defined as P(a, b)=T2[(S(a, b), 1 – S(b, a)], with T 2 being a t-norm. The choice set is given by C3 ( A, S ) a

A

C3 ( a )

max C3 (b) b A (33)

144 A. Bufardi et al. The strict nondomination of a by all other alternatives is defined as follows: C4 ( a ) 1 b A \{ a} T [1 P (b, a )] (34) C4 (a) represents the degree of strict nondomination of all the other alternativ es on a. The choice set is given by C4 ( A, S ) a A C4 ( a ) max C4 (b) b A (35) In contrast to the measurement of the strengths of alternatives, it is also inte resting to measure their weaknesses. Four weakness-based choice functions C5 , C 6 , C7 , and C8 are presented in the following. The weak domination of all alter natives on alternative a, representing the degree to which a is weakly dominated by all the other alternatives, is defined as follows: C5 (a ) 1 b A \{ a } T S (b, a ) (36) The choice set corresponding to the weak domination (of all alternatives on a gi ven alternative) function C5 is given by C5 ( A, S ) a A C5 ( a ) max C5 ( a ) b A (37) The weak nondomination of an alternative a over all other alternatives represent ing the degree to which a doesn’t weakly dominate all the other alternatives is de fined as follows: C6 ( a ) 1 b A \{ a } T [1 S ( a, b)] (38) The choice set corresponding to the weak nondomination (of an alternative on all others) function C6 is given by C6 ( A, S ) a A C6 ( a ) max C6 ( a ) b A

(39)

Fuzzy Outranking Methods 145 The strict domination of all alternatives on alternative a that gives the degree to which a is strictly dominated by all the other alternatives is defined as fo llows: C7 ( a ) 1 b A \{ a } T P (b, a ) (40) The choice set corresponding to the strict domination (of all alternatives on a given alternative) function C7 is given by C7 ( A, S ) a A C7 ( a ) max C7 (b) b A (41) The strict nondomination of an alternative a over all other alternatives represe nting the degree to which a doesn’t strictly dominate all the other alternatives i s defined as follows: C8 (a ) 1 b A \{ a} T [1 P (a, b)] (42) The choice set corresponding to the strict nondomination (of an alternative on a ll others) function C8 is given by C8 ( A, S ) a A C8 (a ) max C8 (b) b A (43) A ranking method can be obtained using the core concept. Once the set of best al ternatives (Ck+1) is chosen by the choice function C(R, Ak), which can be any of the choice functions defined above, it is removed from the initial set A, and a nother core set is found between the remaining alternatives (Ak\Ck+1). This reas oning is applied until the current set (Ak) is empty. This algorithm was propose d in (Perny, 1992), and it is described as follows: Set k := 0 and Ak := A do Wh ile Ak Begin Ck+1 := C(R,Ak) Ak+1 := Ak\ Ck+1 k := k+1 End

146 A. Bufardi et al. R is one of the relations used to define the first four choice functions ( C1 to C4 ), specifically the weak preference S and strict preference P relations. The resulting preorder R is a complete ranking of sets of single or multiple (indif ferent) alternatives from best to worst, where R stands for S or P. Four ranking s can be obtained using the strength concept. The same algorithm can be used to obtain a second type of preorder but this time using the last four functions ( C 5 to C8 ). As they are based on the weakness concept, an ascending preorder from worst to the best will be constructed, denoted by R. These second type of ranki ngs can be different from the previous one. The notions of ascending–descending an d weak–strict rankings are introduced as follows. Similar concepts were used in me thods like ELECTRE II and III, MAPPACC, and PRAGMA. Methods like PROMETHEE I and II use concepts of weakness and strength of alternatives but in a different man ner. Four different preorders can be defined as follows: Descending weak preorde r is the complete ranking obtained using the iterated choice functions C1 or C2 , Descending strict preorder is the complete ranking obtained using the iterated choice functions C3 or C4 , Ascending weak preorder is the complete ranking obt ained using the iterated choice functions C5 or C6 , Ascending strict preorder i s the complete ranking obtained using the iterated choice functions C7 or C8 . L ooking at the choice functions considered, we see that in fact, C5 , C6 , C7 , a nd C8 are “dual” of the functions of C1 , C2 , C3 and C4 respectively. So each pair C1 C5 , C2 C6 , C3 C7 and C4 C8 express the force and the weakness, when used in a ranking procedure. At the same time, pairs like C1 C2 and C3 C4 respectively, C5 C6 and C7 C8 express another type of “duality” that notions of “outgoing” domination–n on domination (i.e., of an alternative on all the other alternatives), when talk ing about strength, respectively “incoming” domination–non domination (i.e., of all al ternatives on the alternative under consideration), when considering the weaknes s. And

Fuzzy Outranking Methods 147 finally, both “dualities” are present for both weak and strict preference relations. The eight functions can be used alone to obtain a final ranking (weak or strict preorder). Nevertheless, the rankings obtained from two preorders (one descendi ng and the other ascending), thus allowing incomparability (since an alternative ai may be preferred over another alternative aj in one preorder and aj preferre d over ai in the other preorder), are richer and more interesting, as they take into account concepts that may be opposite, or dual, as shown above. Various ran king procedures based on a pair of choice functions, together with their charact erization from the following points of view, can be obtained: Type of preference : weak–strict, Type of the ranking of individual choice functions: ascending desce nding, Concept involved: strength–weakness, Intuitive meaning of the individual ch oice functions: incoming domination, incoming nondomination, outgoing domination , and outgoing nondomination. 4.1.4 Illustrative Example The example is adapted from (Wang, 2001). Let us consider the seven valve types (a1 to a7), and the criteria are cost, maintenance, criteria sensitivity, leakag e, rangibility, and stability (g1 to g6). The performance matrix is given in Tab le 5. Table 5. Performance Matrix for Seven Valve Types (Trapezoidal Fuzzy Numbers) Alternatives Criteria’s weights of importance 0.217 0.174 0.174 0.217 0.087 Perfor mance with respect to criterion gk g2 g3 g4 g5 g1 (4, 5, 5, 6) (5, 6, 7, 8) (7, 8, 8, 9) (7, 8, 8, 9) (7, 8, 8, 9) A1 A2 (7, 8, 8, 9) (8, 9, 10, 10) (7, 8, 8, 9 ) (2, 3, 4, 5) (8, 9, 10, 10) (7, 8, 8, 9) (1, 2, 2, 3) (7, 8, 8, 9) (7, 8, 8, 9 ) (5, 6, 8, 9) A3 (1, 2, 4, 5) (4, 5, 5, 6) (4, 5, 5, 6) (2, 3, 7, 8) (4, 5, 8, 9) A4 A5 (7, 8, 8, 9) (5, 6, 7, 8) (5, 6, 7, 8) (8, 9, 10, 10) (1, 2, 2, 3) (4, 5, 5, 6) (4, 5, 5, 6) (2, 3, 4, 5) (5, 6, 7, 8) (8, 9, 10, 10) A6 (4, 5, 7, 8) ( 8, 9, 10, 10) (7, 8, 8, 9) (5, 6, 7, 8) (8, 9, 10, 10) A7 0.131 g6 (1, 2, 2, 3) (7, 8, 8, 9) (5, 6, 7, 8) (8, 9, 10, 10) (1, 2, 2, 3) (8, 9, 10, 10) (7, 8, 8, 9)

148 A. Bufardi et al. The single criterion fuzzy outranking relations Sk are first calculated for each criterion gk, k = 1…6 using relation (25) for a number of -cuts N = 50. In the se cond step, Sk are aggregated using the weighted arithmetic (a particular case of the Choquet intergral) mean with the criteria weights of importance given in Ta ble 5. These steps are repeated for the five representative situations given by the pair of parameters ( , ), representing the decision maker’s attitude. Figures 9 13 represent the above-mentioned situations in terms of the global fuzzy outra nking relation S. 1 1 2 3 4 5 6 7 0.783 2 3 4 5 6 7 0.63 1 2 3 4 5 6 7 1 0.783 0.826 2 3 4 5 6 7 1 0.413 0.652 0.869 0.773 0.638 0.62 0.804 1 0.411 0.652 0.869 0.615 0.802 0.495 1 0.783 0.776 0.783 0.682 0.783 0.66 1 0.6 95 0.658 0.628 1 0.272 0.87 0.245 0.66 0.77 0.392 1 0.519 1 c 1 0.783 0.902 0.783 0.685 0.783 1 0.695 0.663 0.618 0.638 1 0.306 0.605 0.257 1 0.782 0.461 1 0.652 1 c 0.241 0.356 0.388 0.354 0.368 0.446 0.516 0.435 0.446 0.766 0.461 0.635 0.782 0.783 0.478 0.652 0.782 0.837 0.837 0.675 0.899 1 0.782 0.478 0.62 0.899 0.512 0.435 0.426 0.853 0.262 0.817 0.808 0.624 0.902 0.591 0.902 Figure 9. S(ai,aj) for = 0, (conserv-pessim) 1 1 2 3 4 5 6 7 0.783 2 3 4 5 = 6 Figure 10. S(ai,aj) for = 1, (conserv-optim) 7 1 2 3 4 5 6 7 1 0.783 0.164 0.745 2 3 4 5 6 = 7 1 0.404 0.652 0.869 0.598 0.795 0.546 1 0.783 0.822 0.783 0.671 0.783 1 0.695 0. 641 0.617 0.633 1 0.261 0.662 0.305 1 0.782 0.456 0.24 1 0.577 1 m 1 0.396 0.652 0.869 0.578 0.787 0.613 1 0.783 0.876 0.783 0.659 0.783 1 0.695 0. 621 0.61 0.614 0.44 1 a

0.796 0.633 0.761 0.456 0.759 0.614 0.252 0.356 0.402 0.35 0.371 1 0.238 0.562 0.214 1 0.782 1 0.652 0.63 0.782 0.44 0.614 0.782 0.484 0.435 0.419 0.848 0.81 0.807 0.617 0.888 0.452 0.435 0.399 0.832 0.228 0.59 0.888 0.79 0.788 0.578 0.876 0.571 0.876 Figure 11. S(ai, aj) for = 0, (moderate) 1 1 2 3 4 5 6 7 0.783 0.826 = 2 Figure 12. S(ai, aj) for = 0, (agress-pessim) 3 4 5 6 7 = 1 0.395 0.652 0.869 0.576 0.786 0.444 1 0.783 0.734 0.783 0.658 0.783 0.62 1 0.6 95 0.619 0.612 1 0.62 0.23 0.712 0.357 1 0.782 0.444 1 0.484 1 0.248 0.352 0.404 0.749 0.444 0.618 0.782 0.453 0.435 0.404 0.836 0.224 0.795 0.795 0.591 0.875 0.578 0.875 Figure 13. S(ai, aj) for = 1, = a (agress-optim)

Fuzzy Outranking Methods 149 The complete preorders given by the functions C1 ( C4 ), C2 ( C3 ), C5 ( C8 ), a nd C6 ( C7 ) were determined for each of the five representative decision attitu des. Because the fuzzy numbers expressing the performances of the considered alt ernatives interfere very little and in a trivial manner, we observed an influenc e that is not strong enough to change the partial preorders when sliding from a conservative to an aggressive attitude. Some changes are noticed when varying th e other parameter ( ). Table 6 shows the complete preorders: Table 6. Complete Preorder for the Seven Types of Valves No. 1 2 3 4 Choice func tions Decision strategy ( , ) (0, a), (0, c) (1, a), (0.5, m), (1, c) (0, a), (0 , c) (1, a), (0.5, m), (1, c) (0, a), (0, c) (1, a), (0.5, m), (1, c) (0, a), (0 , c) (1, a), (0.5, m), (1, c) Preference 2>3>1>7>5>6>4 2>3>7>5>1>6>4 3, 5, 7 > 2 > 1 > 6 > 4 3, 5, 7 > 2 > 1 > 6 > 4 7>2>5>3>1>6>4 7>2>5>3>1>6>4 7>2>3>1>5>6>4 7 >2>3>5>1>6>4 C1 C4 C 2 C3 C5 C8 C6 C7 For each decision strategy, six partial preorders can be derived from the above table by intersecting pairs of choice functions. They are shown in Table 7. Besi des the theoretical foundations of this method, its advantages are related to th e practical aspects, namely the format of the input data that can be used (gener al, nonanalytical representations of fuzzy numbers) where the preference functio n is described as a vector of -cuts. Six rankings are proposed. One, several, or all of them can be used to reinforce the choice or the ranking. They enclose di fferent choice ideas, all together offering a large “palette” of concepts. It is up to the decision maker which of them is to be used in the concrete problem. The c oncepts that are proposed are easy to understand, and they give transparency to the decision process.

150 Table 7. Partial Ranking of the Eight Valve Types No. Choice functions ( C1 1 ( C2 C3 ) (1, a), (0.5, (1, c) m A. Bufardi et al. Decision strategy ( , ) a2 Preference C4 ) (0, a ), (0, c ) ), a3 a3 a7 a1 a5 a6 a4 a7 a5 a2 a1 a6 a4 ( C1 2 ( C5 C4 ) C8 ) (0, a ), (0, c ) ), a7 a2

a3 a5 a1 a6 a4 (1, a), (0.5, (1, c) m a7 a2 a3 a5 a1 a6 a4 a7 ( C5 3 ( C6 C8 ) C7 ) (0, a ), (0, c ) ), a7 a2 a2 a3 a5 a1 a6 a4 (1, a), (0.5, (1, c) m a5 a3 a2 a1 a6

a4 (0, ( C2 4 ( C6 C7 ) C3 ) a ), (0, c ) a7 a3 a1 a5 a6 a4 (1, a), (0.5, (1, c) m a7 a2 a5 a3 a1 a6 a4 ), ( C1 5 ( C6 C4 ) C7 ) (0, a ), (0, c ) ), a2 a2 a3 a1 a7 a5 a6 a4 (1, a), (0.5, (1, c)

m a3 a7 a5 a1 a6 a4 ( C2 6 ( C5 C3 ) C8 ) (0, a ), (0, c ) ), a7 a5 a2 a3 a1 a6 a4 (1, a), (0.5, (1, c) m a7 a5 a2 a3 a1 a6 a4

Fuzzy Outranking Methods 151 4.2 Other Fuzzy Outranking Methods All outranking methods briefly described in this subsection consider fuzzy evalu ations of alternatives on criteria; therefore, they are fuzzy outranking methods . 4.2.1 Method of Czy ak and S owi ski (1996) This method is an adaptation of ELECTRE III to the case where the concordance an d discordance indices are determined from the fuzzy evaluations of alternatives on criteria through the use of four different measures using possibility and nec essity concepts from possibility theory developed in Dubois and Prade (1988). Th e aggregation of the possibility and necessity measures to drive the concordance and discordance indices is realized through the use of a weighted root-power me an. Apart from an adjustment of the monocriterion concordance and discordance in dices through some transformation, the rest of the method is similar to ELECTRE III. The method is illustrated through its application to the ground water manag ement problem considered in Duckstein et al. (1994). 4.2.2 Method of Wang (1997) This method is based on the consideration of a fuzzy preference relation P defin ed each pair of alternatives (a, b) whose respective fuzzy scores on a given cri terion are A and B as follows: P ( a, b) D( A, B) D( A B,0) D( A, o) D( B,0) (44) where D(A,B) represents the areas where A dominates B, D(A B,0) represents the i ntersection areas of A and B, D(A,0) represents the area of A, and D(B, 0) repre sents the area of B. It is worth recalling that this fuzzy relation is considere d by Tseng and Klein (1989) for the problem of ranking fuzzy numbers. The outran king relation is defined for: The case of a pseudo-order preference model where a preference and indifference thresholds are associated with criteria, The case of a semi-order preference model where only indifference thresholds are associat ed with criteria,

152 A. Bufardi et al. The case of a complete-preorder preference model where the preference and indiff erence thresholds are null for each criterion. The exploitation phase is based o n the consideration of the concepts of dominance and non-dominance sets. The met hod is illustrated through its application to the problem of evaluating and comp aring design concepts in conceptual design. Güngör and Arikan (2000) applied a simil ar method to the problem of energy policy planning. 4.2.3 Method of Wang (1999) In this method, the concordance and discordance indices are determined from the fuzzy evaluations of alternatives on criteria through the use of possibility and necessity measures. More specifically, for two design requirements ri and rj, t he concordance and discordance indices of criterion Ck with the assertion “ri is a t least as good as rj” are defined as follows: CI k (ri , rj ) POSS k (ri DI k (ri , rj ) rj ) (1 NESS k (rj ) NESS k (ri ri ) rj ) (45) (46) 1. where is a preference ratio such that 0 The global outranking relation is obt ained from monocriterion concordance and discordance indices through the use of the aggregation method developed by Siskos et al. (1984). The method is illustra ted through its application to the problem of prioritizing design requirements i n quality function deployment in the case of a car design. 4.2.4 Method of Wang (2001) In this method, the construction of the fuzzy outranking relation is similar to that of Czy ak and S owi ski (1996) since the concordance and discordance indice s are determined from the fuzzy evaluations of alternatives on criteria through the use of four different measures using possibility and necessity concepts. How ever, the exploitation of the global fuzzy outranking relation is different from ELECTRE III since it is based on the determination of the set of nondominated a lternatives as it is considered in Orlovsky (1978). The method is illustrated th rough its

Fuzzy Outranking Methods 153 application to the problem of ranking engineering design concepts in conceptual design. 5. CONCLUSION In this chapter we made a clear distinction between outranking methods based on the construction and exploitation of a valued outranking relation and outranking methods based on the construction and exploitation of a fuzzy outranking relati on since they are applicable to two different situations. Indeed, the outranking methods with a valued outranking relation are applicable to the situation where the evaluations of alternatives on criteria are crisp, whereas the outranking m ethods with a fuzzy outranking relation are applicable to the situation where th e evaluations of alternatives on criteria are fuzzy. The outranking methods with a valued outranking relation are called valued outranking methods and the outra nking methods with a fuzzy outranking relation are called fuzzy outranking metho ds. In the literature the fuzzy and valued outranking methods are often confused and the clarification made in this chapter allows avoiding this confusion. All fuzzy outranking methods deal with the problem of comparing fuzzy numbers; howev er, they consider different approaches: Gheorghe et al. (2004) consider an appro ach based on -cuts; Czy ak and S owi ski (1996) and Wang (1999, 2001) consider a n approach based on possibility and necessity measures; Wang (1997) and Güngör and A rikan (2000) consider an approach based on the comparison of areas of fuzzy numb ers. The valued outranking methods ELECTRE III and PROMETHEE are widely applied to real-world problems; however, they are not suitable to the problems where the evaluations of alternatives on criteria are fuzzy. The fuzzy outranking methods presented in this chapter are quite recent compared with the valued outranking methods, and even if they were applied to specific problems, they can be adapted to any MCDA problem where the evaluations of alternatives on criteria are fuzzy . In this chapter, we provided two illustrative examples, one for a valued outra nking method, namely ELECTRE III, and one for a fuzzy outranking method, namely the method developed by the authors. The objective is to show that outranking me thods can be applied to various problems with the mention that valued outranking methods are suitable to problems with

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FUZZY MULTI-CRITERIA EVALUATION OF INDUSTRIAL ROBOTIC SYSTEMS USING TOPSIS Cengiz Kahraman, Ihsan Kaya, Sezi Çevik, Nüfer Yasin Ates, and Murat Gülbay Istanbul Technical University, Faculty of Management, Department of Industrial E ngineering, Macka Istanbul Turkey Abstract: Industrial robots have been increasingly used by many manufacturing firms in dif ferent industries. Although the number of robot manufacturers is also increasing with many alternative ranges of robots, potential end users are faced with many options in both technical and economical factors in the evaluation of the indus trial robotic systems. Industrial robotic system selection is a complex problem, in which many qualitative attributes must be considered. These kinds of attribu tes make the evaluation process hard and vague. The hierarchical structure is a good approach to describing a complicated system. This chapter proposes a fuzzy hierarchical technique for order preference by similarity ideal solution (TOPSIS ) model for the multi-criteria evaluation of the industrial robotic systems. An application is presented with some sensitivity analyses by changing the critical parameters. Fuzzy sets, TOPSIS, robotic systems, multi-criteria, hierarchy Key words: 1. INTRODUCTION Robotics is the science and technology of robots, their design, manufacturing an d application. Robotics requires a working knowledge of electronics, mechanics, and software, and it is usually accompanied by a large working knowledge of many other subjects. A robot is a mechanical or virtual, artificial agent. It is usu ally an electromechanical system, which, C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 159

160 C. Kahraman et al. by its appearance or movements, conveys a sense that it has intent or agency of its own. The word “robot” can refer to both physical robots and virtual software age nts, but the latter are usually referred to as “bots” to differentiate. While there is still discussion about which machines qualify as robots, a typical robot will have several, though not necessarily all of, the following properties (Craig, 2 005; Tsai, 1999): is not “natural” (i.e., artificially created), can sense its envir onment and manipulate or interact with things in it, has some degree of intellig ence or ability to make choices based on the environment, or has an automatic co ntrol/preprogrammed sequence is programmable, moves with one or more axes of rot ation or translation, makes dexterous coordinated movements, appears to have int ent or agency. An industrial robot is officially defined by ISO (Anonymous, 2007 ) as an automatically controlled, reprogrammable, multipurpose manipulator progr ammable in three or more axes. The field of robotics may be more practically def ined as the study, design, and use of robot systems for manufacturing (a top-lev el definition relying on the prior definition of a robot). During the last decad es, both the ranges of applications and the number of available industrial robot s have substantially increased. Industrial robots are used for many applications , such as assembly, loading and unloading, material handling, spray painting, an d welding. While evaluating industrial robots, potential end users are faced wit h many options in the selection of an appropriate industrial robot to meet their requirements. The decision in robot selection is therefore more complex because many technical and economical factors affect the performance of the industrial robots. Hence, a multi-criteria evaluation approach is required. Fuzzy sets and systems methodologies are useful for modeling uncertainty and imprecision due to the complexity of contemporary industrial robots, which integrate economical an d technical evaluation factors. Humans are unsuccessful in making quantitative p redictions, whereas they are comparatively efficient in qualitative forecasting. Furthermore, humans are more prone to interfere with biasing tendencies if they are forced to provide numerical estimates since the elicitation of numerical es timates forces an individual to operate in a mode that requires more mental effo rt than that required for less precise verbal statements (Karwowski and Mital, 1 986). Since fuzzy linguistic models permit the translation of verbal expressions into numerical ones, thereby dealing

Evaluation of Industrial Robotic Systems 161 quantitatively with imprecision in the expression of the importance of each crit erion, some multi-attribute methods based on fuzzy relations can be used. Applic ations of fuzzy sets within the field of decision making have, for the most part , consisted of extensions or fuzzifications of the classic theories of decisionmaking. Although decision making under conditions of risk and uncertainty has be en modeled by probabilistic decision theories and by game theories, the fuzzy de cision theory attempts to deal with the vagueness or fuzziness inherent in subje ctive or imprecise determinations of preferences, constraints, and goals (Yager, 1982). Several models have been suggested for the robot selection in the past. They can be classified into five categories: (1) multi-criteria decision making (MCDM) models, (2) production system performance optimization models, (3) comput er assisted models, (4) statistical models, and (5) other approaches. MCDM model s include multi-attribute decision making (MADM) models (Agrawal et al., 1991; J ones et al., 1985; Nnaji and Yannacopoulou, 1988) multi-objective decision makin g (MODM) models (Agrawal et al., 1991), and other similar approaches (Huang and Ghandforoush, 1984; Nnaji, 1988). In MADM, all objectives of the decision maker are unified under a superfunction termed the decision maker’s utility, which depen ds on robot attributes. In MODM, the decision maker’s objective, such as optimal u tilization of resources and improved quality, remain explicit and are assigned w eights reflecting their relative importance. The main advantage of MCDM models i s their ability to consider a large number of robot attributes. Using MCDM, the decision maker can consider engineering, vendor-related, and cost attributes; ho wever, for a problem as complex as robot selection, the data requirements these models place on the decision maker may be overwhelming (Narasimhan and Vickery, 1988). Production system performance optimization models select a robot that opt imizes some performance measures of the production system, such as quality or th roughput, with robot attributes treated as decision variables. Computer assisted models have been advocated by many researchers to deal with the large number of robot attributes and available robots (Boubekri et al., 1991; Fisher and Maimon , 1988). In general, the decision maker starts by providing the data about the r obot application. The data are used by an expert system to provide a list of imp ortant robot engineering attributes and their desired values, which in turn is u sed to obtain a list of feasible robots from a descriptive database of available robots. Statistical models focus on the trade-off between robot engineering att ributes and identify robots that provide the best combination of attribute value s. Other approaches to the problem include the development of robot time and mot ion system studies

162 C. Kahraman et al. (Nof, 1985), economic cost/benefit analysis (Nof and Lecthman, 1982) and data en velopment analysis (Knott and Getto, 1982). In this chapter, fuzzy multi-attribu te industrial robotic system selection problem is handled. A fuzzy hierarchical fuzzy Technique for Order Preference by Similarity Ideal Solution (TOPSIS) metho d is developed to solve this multi-attribute selection problem. In the current l iterature, the only method that takes the hierarchy among attributes and alterna tives into consideration is the analytical hierarchy process (AHP). The develope d method, fuzzy hierarchical TOPSIS, also has the ability of considering the hie rarchy among attributes and alternatives. This chapter is organized as follows. In Section 2, the criteria for evaluation of industrial robotic systems are pres ented. The fuzzy multicriteria hierarchical TOPSIS method for industrial robot s election is developed in Section 3. Evaluation of industrial robotic systems is illustrated in Section 4. Finally, concluding remarks are given in Section 5. 2. CRITERIA FOR EVALUATION OF INDUSTRIAL ROBOTIC SYSTEMS Traditional economic analysis techniques incorporate direct costs (and benefits) to which dollar values can be attached. Using these techniques, the evaluation of robots may result in an expected loss or negative return. Management must acc urately assess the value of the intangible benefits provided by the investment i n automation against the cost figures. If those responsible for the decision and the commitment of company resources do consider the intangible benefits (precis ion or accuracy, programmability, etc.) to be greater than the cost, then an inv estment in robots is justified. Thus, evaluation of the industrial robotic syste ms is often specified using many parameters that can be categorized into two mai n groups: 1. Economical Attributes Investment Costs (InvC) o Purchase Cost (PC): The basic costs of planning and design by the user company’s engineering staff to install the robot. o Installation Cost (InsC): This includes the labor and mate rials needed to prepare the installation site. o Special Tooling Cost (ST): This includes the cost of the the fixtures and tools required to operate the work ce ll.

Evaluation of Industrial Robotic Systems 163 Operating Costs (OC) o Maintenance Cost (MaC): This covers the anticipated costs of maintenance and repair for the robot cell. o Labor Cost (LC): The direct lab or costs associated with the operation of the robot cell and the indirect labor costs that can be directly allocated to the operation of the robot cell. o Train ing Cost (TC): Training is a continuing activity in which much of it is required for the installation. 2. Technical Attributes Repeatability (Rep): This is a me asure of the ability of the robot to return to its original point. Speed (Sp) Me mory capacity (MeC) Precision or accuracy (Pre) Programmability (Pro) Number of axes (NA) Workload (Wl) The hierarchy considered in the study is given in Figure 1. Selection of the best industrial robotic systems Economical Technical InvC OC Sp Wl Pro NA Rep Pre MeC PC InsC ST MaC LC TC AH1 AH2 AH3 Figure 1. The hierarchy to evaluate industrial robotic systems

164 C. Kahraman et al. 3. FUZZY MULTI-ATTRIBUTE DECISIONMAKING METHODS Fuzzy sets were introduced by Zadeh in 1965 to represent/manipulate data and inf ormation possessing nonstatistical uncertainties (Zadeh, 1965). It was specifica lly designed to represent mathematical uncertainty and vagueness and to provide formalized tools for dealing with the imprecision intrinsic to many problems. Fu zzy logic provides an inference morphology that enables approximate human reason ing capabilities to be applied to knowledge-based systems. The theory of fuzzy l ogic provides a mathematical strength to capture the uncertainties associated wi th human cognitive processes, such as thinking and reasoning. Some essential cha racteristics of fuzzy logic are related to the following: Exact reasoning is vie wed as a limiting case of approximate reasoning, everything is a matter of degre e, knowledge is interpreted a collection of elastic or, equivalently, fuzzy cons traint on a collection of variables, inference is viewed as a process of propaga tion of elastic constraints, and any logical system can be fuzzified. Two main c haracteristics of fuzzy systems give them better performance for specific applic ations: Fuzzy systems are suitable for approximate reasoning, especially for the system with a mathematical model that is difficult to derive, and fuzzy logic a llows decision making with estimated values under incomplete or uncertain inform ation. Fuzzy multi-criteria decision making (FMCDM) has provoked great interest in decision science, systems engineering, management science, and operations res earch. Fuzzy multi-attribute decision making is an important component of the FM CDM. Many efficient methods for fuzzy multiattribute decision making problems ex ist with the decision maker’s preference information completely known and complete ly unknown. The key to solving fuzzy multi-criteria decision making problems is how to obtain preference information of the decision-maker, i.e., criteria weigh ts. Many efficient methods have been presented for the fuzzy multicriteria decis ion making problems with the decision maker’s preference information completely kn own and completely unknown, such as, TOPSIS method, AHP, average weighted compre hensive method, fuzzy optimum seeking method, minimum membership degree method, average weighted programming method, fuzzy neural networks comprehensive decisio n making method, fuzzy iteration method, and target decision by entropy weight a nd fuzzy. But, no research exists in fuzzy multi-criteria decision making situat ed between the above extreme circumstances, i.e., the fuzzy

Evaluation of Industrial Robotic Systems 165 multi-criteria decision making with incomplete information. Therefore, research of such problems is of importance to scientific research and real applications. 3.1 Fuzzy TOPSIS TOPSIS views a MADM problem with m alternatives as a geometric system with m poi nts in the n-dimensional space. It was developed by Hwang and Yoon (1981). The m ethod is based on the concept that the chosen alternative should have the shorte st distance from the positive-ideal solution and the longest distance from the n egative-ideal solution. TOPSIS defines an index called similarity (or relative c loseness) to the positiveideal solution and the remoteness from the negative-ide al solution. Then the method chooses an alternative with the maximum similarity to the positive-ideal solution. Using the vector normalization, the method choos es the alternative with * the largest value of Ci as given in Eq. 1. 2 n wj j 1 xij m 2 xij i 1 2 vj (1) 2 Ci* n wj j 1 xij m n v* j x 2 ij j 1 wj xij m vj x 2 ij i 1 i 1 or it chooses the alternative with the least value of Ci formulated as in Eq. 2.

2 n wj j 1 xij m v* j x 2 ij 2 n (2) Ci n i 1 2 wj j 1 xij m v* j x 2 ij j 1 wj xij m vj x 2 ij i 1 i 1

166 C. Kahraman et al. where i (i = 1 ,…, m) and j (j = 1 ,…, n) are index numbers for the alternatives and attributes, respectively; w j is the weight of the jth attribute; xij is the at tribute rating for ith alternative’s jth attribute; v * is j the positive-ideal va lue for jth attribute, where it is a maximum for benefit attributes and a minimu m for cost attributes; and v j is the negative-ideal value for the jth attribute , where it is a minimum for benefit attributes and a maximum for cost attributes . In the last decade, some fuzzy TOPSIS methods were developed in the literature : Chen and Hwang (1992) transform Hwang and Yoon’s (1981) method into a fuzzy case . Liang (1991) presents a fuzzy multi-criteria decision making based on the conc epts of ideal and anti-ideal points. The concepts of fuzzy set theory and hierar chical structure analysis are used to develop a weighted suitability decision ma trix to evaluate the weighted suitability of different alternatives versus crite ria. Triantaphyllou and Lin (1996) develop a fuzzy version of the TOPSIS method based on fuzzy arithmetic operations, which leads to a fuzzy relative closeness for each alternative. This fuzzy TOPSIS method offers a fuzzy relative closeness for each alternative; the closeness is badly distorted and over-exaggerated bec ause of the reason of fuzzy arithmetic operations. Chen (2000) describes the rat ing of each alternative and the weight of each criterion by linguistic terms, wh ich can be expressed in triangular fuzzy numbers. Then, a vertex method for TOPS IS is proposed to calculate the distance between two triangular fuzzy numbers. C heng et al. (2002) apply Chen and Hwang’s (1992) fuzzy TOPSIS approach for solving the solid waste management problem in Regina of Saskatchewan Canada. Additional ly, they apply four other MCDM methods for the analysis of solid-waste managemen t systems, including simple weighted addition (SWA) method, weighted product (WP ) method, cooperative game theory, and ELECTRE. Since all methods result in diff erent rankings of the alternative solutions, they use an aggregation approach ca lled the average ranking procedure to analyze the results. Zhang and Lu (2003) p resent an integrated fuzzy group decision making method in order to deal with th e fuzziness of preferences of the decision-makers. In this chapter, the weights of the criteria are crisp values gathered by pair-wise comparisons where the pre ferences of the decision makers are represented by triangular fuzzy numbers (TFN s). Chen and Tzeng (2004) transform a fuzzy MCDM problem into a nonfuzzy MCDM us ing a fuzzy integral. Instead of using distance, they employ a gray relation gra de to define the relative closeness of each alternative. Abo-Sinna and Abou-ElEn ien (2005) extend the technique for order preference by similarity ideal solutio n (TOPSIS) for solving large-scale multiple objective programming

Evaluation of Industrial Robotic Systems 167 problems involving fuzzy parameters. Wang and Elhag (2006) present a nonlinear p rogramming (NLP) solution procedure using a fuzzy TOPSIS method based on alpha l evel set. They discuss the relationship between the fuzzy TOPSIS method and the fuzzy weighted average (FWA). They illustrate three examples about bridge risk a ssessments to compare the proposed fuzzy TOPSIS and other procedure. Jahanshahlo o et al. (2006) study the cases in which determining precisely the exact value o f the attributes is difficult, and as a result of this, the attribute values sho uld be considered as intervals. They aim to extend the TOPSIS method for decisio n making problems with interval data. By extension of the TOPSIS method, they pr esent an algorithm to determine the most preferable choice among all possible ch oices, when data are interval. Table 1. A Comparison of Fuzzy TOPSIS Methods Source Chen and Hwang (1992) Liang (1999) Chen (2000) Chu (2002) Tsaur et al. (2002) Attribute Weights Fuzzy Numbe rs Fuzzy Numbers Fuzzy Numbers Fuzzy Numbers Crisp Values Type of Fuzzy Ranking Method Numbers Lee and Li’s (1988) Trapezoidal generalized mean method Chen’s (1985) ranking Trapezoidal with maximizing set and minimizing set Chen (2000) proposes Triangular vertex method Liou and Wang’s (1992) Triangular ranking method of tota l integral value with =1/2 Zhao and Govind’s (1991) center of area Triangular meth od Chen’s (2000) vertex Triangular method Kaufmann and Gupta’s Triangular (1988) mea n of the removals method Cha and Yung (2003) Triangular propose a fuzzy distance operator Triangular Triangular Interval data Chen’s (2000) vertex method Normaliz ation Method Linear Normalization Manhattan distance Linear Normalization Modifi ed Manhattan distance Vector Normalization Manhattan distance Linear Normalizati on Linear Normalization Zhang and Lu Crisp Values (2003) Chu and Lin (2003) Cha and Yung (2003) Yang and Hung (2005) Wang and Elhag (2006) Jahanshahloo et al. (2006) Fuzzy Numbers Cris p Values Fuzzy Numbers Fuzzy Number Crisp Values Normalized fuzzy linguistic ratings are used Chen’s (2000) vertex Linear method No rmalization Jahanshahloo et al. (2006) propose a new normalization & ranking met hod

168 C. Kahraman et al. A comparison of the fuzzy TOPSIS methods in the literature is given in Table 1. The comparison includes the computational differences among the methods. In this chapter, we prefer Chen and Hwang’s (1992) fuzzy TOPSIS method since the other fu zzy TOPSIS methods are derived from this method with minor differences. In the f ollowing discussion, the steps of fuzzy TOPSIS developed by Chen and Hwang (1992 ) are given. First, a decision matrix, D, of m n dimension is defined as in Eq. 3. X1 x11 A1 D Ai xi1 Am xm1 Xj x1 j xij xmj Xn x1n xin xmn (3) where xij , i , j may be crisp or fuzzy. If xij is fuzzy, it is represented by a trapezoidal number as xij aij ,bij ,cij ,d ij shown in Figure 2. The fuzzy weig hts can be described by Eq. 4. wj ij , ij , ij , ij (4) (x) xij wj 1.0 .0.5 0.0 j j j j aij bij cij dij Figure 2. Trapezoidal fuzzy numbers 3.2 Algorithm

The problem is solved using the following steps.

Evaluation of Industrial Robotic Systems 169 Step 1. Normalize the Decision Matrix. The decision matrix must first be normali zed so that the elements are unit-free. To avoid the complicated normalization f ormula used in classic TOPSIS, we use linear scale transformation as follows: rij xij x*j , j , x j is a benefit attribute x j xij , j , x j is a cost attribu te (5) By applying Eq. 5, we can rewrite the decision matrix in Eq. 3 as in Eq. 6. X1 A1 r11 D Ai ri1 Am rm1 Xj r1 j rij rmj Xn r1n rin rmn (6) When xij is crisp, its corresponding rij must be crisp; when xij is fuzzy, its c orresponding rij must be fuzzy. Eq. 5 is then replaced by the following fuzzy op erations: Let xij aij ,bij ,cij ,d ij and x j* a j* ,b j* ,c j* ,d j* , we have: xij ( )x* j rij x j ( )xij aij d * j , bij c * j , cij b * j , dij aj * (7) ai bi ci di , , , dij cij bij aij Step 2. Obtain the Weighted Normalized Decision Matrix. This matrix is obtained using vij rij w j , j,j (8) When both rij and wij are crisp, vij is crisp. When either rij or wij (or both) are fuzzy, Eq. 8 may be replaced by the following fuzzy operations:

170 C. Kahraman et al. vij rij (.)w* j aij d * j j , bij c * j j , cij b * j j , dij aj * j (9) vij rij (.) w * j ai d ij bi j, cij cij, bij d ij, aij j (10) Eq. 9 is used when the jth attribute is a benefit attribute. Eq. 10 is used when the jth attribute is a cost attribute. The result of Eqs. 9 and 10 can be summa rized as in Eq. 11. X1 A1 v11 v Ai vi1 Am vm1 Xj v1 j vij vmj Xn v1n vin vmn (11) Step 3. Obtain the Positive Ideal Solution (PIS), A* , and the Negative Ideal So

lution (NIS), A . PIS and NIS are defined as A* A where v* j max vij and v j i * * v1 , … , vn (12) (13) v1 , … , vn min vij . i For crisp data, v*j and v j are obtained in a straight forward manner. In the ca se of fuzzy data, v*j and v j may be obtained through some ranking procedures. C hen and Hwang use Lee and Li’s ranking method for comparison of fuzzy numbers. The v*j and v j are the fuzzy numbers with the largest generalized mean and the sma llest generalized mean, respectively. The generalized mean for fuzzy number vij , i , j , is defined as M ( vij ) 2 aij 2 2 2 bij cij d ij aij bij cij d ij 3 aij bij cij d ij (14)

Evaluation of Industrial Robotic Systems 171 For each column j, we find a vij whose greatest mean is v* and whose j lowest me an is v j . Step 4. Obtain the Separation Measures Si* and Si-. In the classic c ase, seperation measures are defined as: n Si* j 1 D* , i 1,..., m ij (15) and, n Si j 1 Dij , i 1,...,m . (16) * For crisp data, the difference measures Dij and Dij are given as * Dij v ij v * j v ij v j . (17) (18) Dij The computation is straightforward. For fuzzy data, the difference between two f uzzy numbers vij (x ) and v*j ( x ) (based on Zadeh (1965)’s study) is explained a s given in Eq. 19. * D ij 1 sup x v ij ( x)^ v* j (x) 1 L ij , i , j (19)

where Lij is the highest degree of similarity of vij and v*j . The value of Lij is best depicted in Figure 3. Similarly, the difference between vij (x ) and vj (x ) is defined as Dij 1 sup x vij ( x)^ vj ( x) 1 Lij , i, j . (20) * Note that both Dij , Dij are crisp numbers.

172 C. Kahraman et al. Step 5. Compute the Relative Closeness to Ideals. This index is used to combine S i* and S i indices calculated in Step 4. Since S i* and S i are crisp numbers, they can be combined: Ci Si Si* Si (21) The alternatives are ranked in descending order of the Ci index. (x) 1.0 vij vj* Lij 0.0 x Figure 3. The derivation of Lij 3.3 Fuzzy Hierarchical TOPSIS In the literature, one of the most known and widely used multi-attribute decisio n making methods is fuzzy AHP. There are two main differences between AHP and TO PSIS. (1) Pair-wise comparisons for attributes and alternatives are made in AHP, although there is no pair-wise comparison in TOPSIS. (2) AHP uses a hierarchy o f attributes and alternatives, whereas TOPSIS does not. The consideration of the hierarchies in the multiattribute problems provides a great superiority to AHP. The developed fuzzy TOPSIS methods today do not take the hierarchies in the mul tiattribute problems into consideration. In the following discussion, we develop a fuzzy hierarchical TOPSIS to solve multi-attribute hierarchical problems. The fuzzy TOPSIS algorithm considering a hierarchy is developed below. The hierarch y given in Figure 4 will be considered.

Evaluation of Industrial Robotic Systems GOAL 173 MA1 MA2 ... MAp ... MAn SA11 SA12 ... SA1r1 SA21 SA22 ... SA2r2 ... SApl ... SAn1 SAn2 ... SAnrn Figure 4. The hierarchy considered in fuzzy TOPSIS algorithm Assume that we have n main attributes, m sub-attributes, k alternatives, and s r espondents. Each main attribute has ri sub-attributes where the total number of sub-attributes m is equal to the sum of ri , i = 1,2,3,…n. ~ The first matrix ( I MA ), given by Eq. 22, is constructed from the weights of the main attributes wi th respect to the goal. Goal MA1 MA2 I MA w1 w2 MA p w p MAn wn (22)

~ where w p is the arithmetic mean of the weights assigned by the respondents an d is calculated by Eq. 23 s ~ wp ~ w pi s , p 1,2,..., n (23) i 1

174 C. Kahraman et al. ~ where w pi denotes the fuzzy evaluation score of the pth main attribute with ~ respect to the goal assessed by the ith respondent. The second matrix ( I SA ) represents the weights of the sub-attributes with respect to the main ~ ~ attrib utes. The weights vector obtained from I MA is written above this I SA as illust rated in Eq. 24. w2 w1 MA1 MA2 SA11 w11 0 SA12 w12 0 SA1r1 SA21 SA22 I SA SA2 r2 SA pl SAn1 SAn 2 SAnrn w1r1 0 0 0 0 0 0 0 0 w21 w22 w2 r2 0 0 0 0 0 w pl 0 0 0 0 0 w n1 wn 2 wnr n wp MA p 0 0 0 0 0 0 wn MAn 0 0 0 0 0 0 (24) ~ where w pl is the arithmetic mean of the weights assigned by the respondents a nd is calculated by Eq. 25. s ~ wpl ~ wpli s (25) i 1 ~ where w pli is the weight of the lth sub-attribute with respect to the pth mai n ~ attribute assessed by the ith respondent. The third matrix ( I A ) is formed by the scores of the alternatives with respect to the sub-attributes. The weigh ts ~ ~ vector obtained from I SA are written above this I A as in Eq. 26.

Evaluation of Industrial Robotic Systems 175 ~ IA ~ W11 SA11 ~ c A1 ~111 A2 c 211 ~ W12 SA12 ~ c112 ~ c 212 … … … … ~ W1r1 SA1r1 ~ c11r1 ~ c 21r1 … … … … ~ W pl SA pl ~ c1 pl ~ c 2 pl … … … … ~ Wnrn SAnrn ~ c1nrn ~ c 2 nrn ~ ~ ~ ~ ~ Aq c q11 c q12 … c q1r1 … c qpl … c qnrn ~ ~ ~ ~ ~ Ak c k 11 c k 12 … c k 1r … c kpl … c knr 1 n (26) where ~ W pl Since w pj n j 1 ~ ~ w p w pj . (27) 0 for j l , we can use Eq. 28 instead of Eq. 27 ~ W pl ~ ~ ~ w p w pl . (28) ~ In I A , cqpl is the arithmetic mean of the scores assigned by the respondents , and it is calculated by Eq. 29 s ~ c qpl ~ c qpli s (29) i 1

~ where cqpli is the fuzzy evaluation score of the qth alternative with respect to the lth sub-attribute under the pth main attribute assessed by the ith respon dent. To determine the importance degree of each main attribute with respect to the goal and each sub-attribute with respect to the mainattributes, Table 2 is u sed. The linguistic terms represented by TFNs for scoring the alternatives under the sub-attributes are given in Table 3. Table 2. The Importance Degrees Very low Low Medium High Very High (0, 0, 0.2) ( 0, 0.2, 0.4) (0.3, 0.5, 0.7) (0.6, 0.8, 1) (0.8, 1, 1)

176 Table 3. The Scores Very low Low Medium High Very High (0, 0, 20) (0, 20, 40) (3 0, 50, 70) (60, 80, 100) (80, 100, 100) C. Kahraman et al. 4. EVALUATION OF INDUSTRIAL ROBOTIC SYSTEMS Three different industrial robotic systems are evaluated using the multiattribut e decision making technique given above. Taking the hierarchy given in Figure 1 into consideration, a questionnaire for fuzzy TOPSIS was prepared to receive the weights of main, sub, and sub-sub-attributes from the experts. A part of this q uestionnaire is given in Appendix A. The questionnaire is applied to a big autom otive company in Turkey. Twentyfour professionals in a company where 4 of them a re top managers, 8 of them are division managers of related departments, and 12 of them are engineers, were interviewed. The response rate was 100% with a high support of top management. First, equations in the fuzzy TOPSIS algorithm using trapezoidal fuzzy numbers given in Section 3.1 are rewritten for TFNs that are c onsidered in this application. Since a TFN (a, b, c) can be represented in trape zoidal form as (a, b, b, c), it can be easily seen that Eq. 7 can be expressed a s follows: xij ( )x*j rij x j ( )xij Then, Eq. 14 is reduced to Eq. 31. aij d *j ai d ij , bij d ij , b* a*j j b d , i , i bij aij (30) M (vij ) 2 aij 2 d ij aij bij bij d ij 3 aij d ij (31)

Evaluation of Industrial Robotic Systems * Dij and Dij are calculated by the Eqs. 32 and 33, respectively. 177 1 * Dij b * 1 bij cij cij c* c* a* a * bij aij aij b* for bij for b * b* i, j bij (32) 1 Dij 1 bij b c c cij cij aij aij a a b bij for b for bij bij i, j b (33) where v* ( a * , b* , c* ) and v j ( a , b , c ) are the fuzzy numbers with the j largest generalized mean and the smallest generalized mean, respectively. Then , the steps of the hierarchical fuzzy TOPSIS algorithm are executed. Our model h as two main attributes, nine sub-attributes, six subsub-attributes, and three al ternatives. Evaluations from all 24 respondents ~ ~ ~ are taken, and I MA , I SA , and I A are obtained and given in Tables 4–7. Table 4. I MA GOAL (0.54, 0.83, 0.91) (0.32, 0.71, 0.86) Table 5. I SA EA (0.27, 0.69, 0.87) (0.32, 0.61, 0.83) 0 0 0 0 0 0 0 TA 0 0 (0.54, 0.76, 0.81) (0.63, 0 .78, 0.86) (0.45, 0.53, 0.74) (0.16, 0.41, 0.56) (0.21, 0.56, 0.71) (0.35, 0.78, 0.86) (0.41, 0.68, 0.87) ~ EA TA ~ InvC OC SP Wl Pro NA Rep Pre MeC

178 Table 6. I SSA Inv (0.53, 0.78, 0.87) (0.45, 0.53, 0.74) (0.35, 0.42, 0.69) 0 0 0 Table 7. I A AH-1 (17, 39, 56) (45, 78, 89) (23, 38, 48) (10, 21, 36) (34, 51, 63) (13, 42, 67) (24, 54, 81) (16, 31, 46) (46, 69, 81) (15, 32, 43) (32, 44, 5 3) (13, 45, 72) (56, 65, 78) (16, 31, 46) AH-2 (45, 78, 89) (32, 45, 76) (38, 44 , 49) (29, 39, 52) (25, 33, 37) (32, 45, 76) (13, 34, 56) (23, 35, 56) (47, 61, 76) (9, 21, 39) (27, 34, 55) (21, 56, 78) (17, 32, 29) (23, 35, 56) C. Kahraman et al. ~ PC InsC ST Mac LC TC OC 0 0 0 (0.63, 0.78, 0.86) (0.61, 0.88, 0.93) (0.21, 0.56, 0.71) ~ PC InsC STF Mac LC TC SP Wl Pro NA Rep Pre MeC Wl AH-3 (21, 28, 46) (13, 26, 57) (41, 53, 62) (41, 53, 62) (9, 17, 33) (6, 22, 46) (29, 36, 48) (32, 45, 76) (29, 36, 48) (39, 63, 81) (15, 32, 43) (36, 67, 85) ( 44, 72, 87) (32, 45, 76) * The tables to obtain rij , vij , M (vij ) , Dij , and Dij are given in Appendi x B. Table 8 shows the distances from the ideal solution for each AH and the nor malized values which makes the results’ interpretation easier. Table 8. Distances from the Ideal Solution Si* 1.184733 1.644703 1.508918 Si 1.7 91474 1.320108 1.440309 Ci 0.601932 0.445259 0.488368 Normalized Ci 0.39 0.29 0. 32 AH-1 AH-2 AH-3 The results in Table 8 indicate that AH-1 achieves the highest performance, wher eas AH-2 has the lowest. To analyze the attitude of the alternatives under diffe rent main attribute weights, a sensitivity analysis is made. The results of sens itivity analyses are given in Table 9 and Figure 5. In Table 9, the states where one of the

Evaluation of Industrial Robotic Systems 179 main attributes has the maximum weight, whereas the other that has less values g iven in Table 2 are examined. For each state, Normalized Relative Closeness to I deals (Ci) is computed. Figures 5 and 6 illustrate the graphical representation of these results. Table 9. The Results of Sensitivity Analyses Normalized Ci AH-1 AH-2 0.39 0.28 0 .39 0.29 0.38 0.30 0.38 0.32 0.35 0.44 AH-1 AH-2 0.39 0.28 0.39 0.26 0.40 0.26 0 .40 0.23 0.42 0.14 EA 0.8, 1, 1 0.8, 1, 1 0.8, 1, 1 0.8, 1, 1 0.8, 1, 1 EA 0.8, 1, 1 0.6, 0.8, 1 0. 3, 0.5, 0.7 0, 0.2, 0.4 0, 0, 0.2 TA 0.8, 1, 1 0.6, 0.8, 1 0.3, 0.5, 0.7 0, 0.2, 0.4 0, 0, 0.2 TA 0.8, 1, 1 0.8, 1 , 1 0.8, 1, 1 0.8, 1, 1 0.8, 1, 1 States 1 2 3 4 5 States 1 2 3 4 5 AH-3 0.32 0.31 0.30 0.29 0.19 AH-3 0.32 0.33 0.33 0.35 0.42 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 1 2 3 States 4 5 Normalized Ci AH1 AH2 AH3 Figure 5. Sensitivity analysis for the case where EA has the highest weight As shown in Figure 5, the importance of technological attributes decreases, wher eas the score of AH-2 increases dramatically and the score of AH-1 decreases. Th is means AH-2 has superior economical properties. Figure 6 shows that AH-1 gets more superior to the others as importance of economical attributes decreases. Th is is because AH-1 has better scores than the others on the technical attributes .

180 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 C. Kahraman et al. Normalized Ci AH1 AH2 1 2 3 States 4 5 AH3 Figure 6. Sensitivity analysis for the case where TA has the highest weight 5. CONCLUSION In this chapter, a model for evaluating and selecting among industrial robotic s ystems has been presented. The model is based on the premise that industrial rob otic systems selection should be viewed as a product of economical and technical attributes. Economical attributes consist of investment costs and operating cos ts, whereas technical attributes consist of memory capacity, speed, and number o f axes, precision, programmability, repeatability, and workload. In addition, pu rchasing costs, installation costs, and special tooling are the sub-attributes o f investment costs and maintenance. Labor and training costs are the sub-attribu tes of operating costs. Industrial robotic system selection is a complex problem in which many qualitative attributes must be considered. These kinds of attribu tes make the evaluation process hard and vague. The hierarchical structure is a good approach to describe a complicated system. The judgments from experts are a lways vague rather than crisp. It is suitable and flexible to express the judgme nts of experts in fuzzy numbers instead of in crisp numbers. Fuzzy AHP has the c apability of taking pair-wise comparisons of these attributes into account with a hierarchical structure. Many fuzzy TOPSIS methods have been proposed without c onsidering these pair-wise comparisons between attributes and a hierarchical str ucture by today. To be able to benefit from the superiority of a hierarchical st ructure, a hierarchical fuzzy TOPSIS method has been developed. It is clear that the selection of an industrial robotic system is a difficult and sensitive issu e

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Evaluation of Industrial Robotic Systems 183 APPENDIX A: QUESTIONNAIRE FOR FUZZY TOPSIS With respect to the overall goal “Selection of the best industrial robotic systems” Q1. What degree of importance do you assign to the main attribute Economic Attri butes? Q2. What degree of importance do you assign to the main attribute Technic al Attributes? With respect to: Overall goal Attributes Questions Importance of one attribute with respect to overall goal (0.3, 0.5, 0.7) Medium (0, 0.2, 0.4) Low (0.6, 0.8, 1) High (0.8, 1 1) Very High (0.8, 1 1) Very High Q1 Q2 Econ Tech Figure A.1. Questionnaire form used to facilitate importance of main attributes with respect to the overall goal With respect to the main attribute Economic Att ributes Q3. What degree of importance do you assign to the sub-attribute Investm ent Costs (InvC)? Q4. What degree of importance do you assign to the sub-attribu te Operating Costs (OC)? With respect to: Economic Attributes Attributes Questions (0, 0, 0.2) Very Low Importance of one sub-attribute with respect to main attribute Research (0.3, 0.5, 0.7) Medium (0, 0.2, 0.4) Low (0.6, 0.8, 1) High (0, 0, 0.2) Very Low Q3 Q4 InvC OC Figure A. 2. Questionnaire forms used to facilitate importance of sub-attributes with respect to main attributes With respect to the sub-attribute Investment Co sts (InvC) Q5. What degree of importance do you assign to the sub-attribute Spec ial Tooling (ST)? Q6. What degree of importance do you assign to the sub-attribu te Installation costs (InsC)? Q7. What degree of importance do you assign to the sub-attribute Purchase cost (PC)?

184 With respect to: Service Questions SubAttributes C. Kahraman et al. Importance of one sub-attribute with respect to main attribute Service (0.3, 0.5 , 0.7) Medium (0, 0.2, 0.4) Low (0.6, 0.8, 1) High Q5 Q6 Q7 ST InsC PC Figure A.3. Questionnaire forms used to facilitate importance of sub-sub-attribu tes with respect to sub-attributes Scoring of Alternatives with respect to sub-s ub-attributes Q8. What scores do you assign to each Industrial Robotic System wi th respect sub-sub-attribute Installation costs (InsC)? Q9. What scores do you a ssign to each Industrial Robotic System with respect sub-sub-attribute Special T ooling (ST)? Q10. What scores do you assign to each Industrial Robotic System wi th respect sub-sub-attribute Purchase cost (PC)? Q11. What scores do you assign to each Industrial Robotic System with respect sub-sub-attribute Maintenance Cos ts (MaC)? Q12. What scores do you assign to each Industrial Robotic System with respect sub-sub-attribute Labor Costs (LC)? Q13. What scores do you assign to ea ch Industrial Robotic System with respect sub-sub-attribute Training Costs (TC)? High (60, 80, 100) Alternatives Fair (30, 50, 70) Low (0, 20, 40) Very Low (0, 0 , 20) AH-1 Q8 InsC AH-2 AH-3 AH-1 Q9 … Q13 ST … TC AH-2 AH-3 … AH-1 AH-2 AH-3 … … … … … Figure A.4. Questionnaire form used to facilitate scores of alternatives with re spect to suband sub-sub-attributes Very High (80, 100, 100) Attributes Questions (0.8, 1 1) Very High (0, 0, 0.2) Very Low to the to the to the to the to the to the

Evaluation of Industrial Robotic Systems 185 APPENDIX B Table B.1. AH-1 PC InsC STF Mac LC TC SP Wl Pro NA Rep Pre MeC (0.191, 0.5, 1.244) (0.506, 1, 1.978) (0.371, 0.717, 1.171) (0.161, 0.396, 0.878) (0.54, 1, 1.853) (0.171, 0 .933, 2.094) (0.296, 1, 2.793) (0.211, 0.689, 1.438) (0.568, 1, 1.723) (0.185, 0 .508, 1.103) (0.582, 1, 1.656) (0.153, 0.672, 2) (0.644, 0.903, 1.393) rij AH-3 (0.236, 0.359, 1.022) (0.146, 0.333, 1.267) (0.661, 1, 1.512) (0.661, 1, 1. 512) (0.143, 0.333, 0.971) (0.079, 0.489, 1.438) (0.358, 0.667, 1.655) (0.421, 1 , 2.375) (0.358, 0.522, 1.021) (0.481, 1, 2.077) (0.273, 0.727, 1.344) (0.424, 1 , 2.361) (0.506, 1, 1.554) AH-2 (0.506, 1, 1.978) (0.36, 0.577, 1.689) (0.613, 0.83, 1.195) (0.468, 0.736, 1.268) (0.397, 0.647, 1.088) (0.421, 1, 2.375) (0.16, 0.63, 1.931) (0.303, 0.778 , 1.75) (0.58, 0.884, 1.617) (0.111, 0.333, 1) (0.491, 0.773, 1.719) (0.247, 0.8 36, 2.167) (0.195, 0.444, 0.518) Table B.2. AH-1 PC InsC STF Mac LC TC SP Wl Pro NA Rep Pre MeC (0.015, 0.223, 0.857) (0.033 , 0.304, 1.159) (0.019, 0.172, 0.64) (0.018, 0.156, 0.57) (0.057, 0.446, 1.302) (0.006, 0.265, 1.123) (0.051, 0.54, 1.946) (0.042, 0.382, 1.063) (0.082, 0.376, 1.097) (0.009, 0.148, 0.531) (0.039, 0.398, 1.011) (0.017, 0.372, 1.479) (0.084, 0.436, 1.042) vij AH-3 (0.018, 0.16, 0.704) (0.01, 0.101, 0.742) (0.034, 0.241, 0.826) (0.072, 0.3 95, 0.982) (0.015, 0.149, 0.682) (0.003, 0.139, 0.771) (0.062, 0.36, 1.153) (0.0 85, 0.554, 1.757) (0.052, 0.196, 0.65) (0.025, 0.291, 1) (0.018, 0.289, 0.82) (0 .047, 0.554, 1.746) (0.066, 0.483, 1.162) AH-2 (0.039, 0.447, 1.362) (0.024, 0.175, 0.989) (0.031, 0.2, 0.653) (0.051, 0.2 91, 0.824) (0.042, 0.288, 0.764) (0.015, 0.284, 1.274) (0.028, 0.34, 1.345) (0.0 61, 0.431, 1.294) (0.084, 0.333, 1.029) (0.006, 0.097, 0.482) (0.033, 0.307, 1.0 49) (0.028, 0.463, 1.602) (0.026, 0.215, 0.387)

186 Table B.3. AH-1 PC InsC STF Mac LC TC SP Wl Pro NA Rep Pre MeC 0.365 0.498 0.276 0.248 0.60 1 0.464 0.845 0.495 0.518 0.229 0.482 0.622 0.5208158 C. Kahraman et al. M (vij ) AH-2 0.616 0.396 0.294 0.388 0.364 0.524 0.570 0.595 0.481 0.194 0.463 0.697 0.2 09 * Dij AH-3 0.294 0.284 0.366 0.483 0.281 0.304 0.524 0.798 0.299 0.438 0.375 0.782 0.5 70 Table B.4. AH-1 PC InsC STF Mac LC TC SP Wl Pro NA Rep Pre MeC 0.214 0.000 0.101 0.323 0.00 0 0.0167 0.000 0.149 0.000 0.220 0.000 0.112 0.045 AH-2 0.000 0.118 0.061 0.121 0.181 0.000 0.133 0.092 0.044 0.298 0.082 0.055 0.4 55 AH-3 0.300 0.222 0.000 0.000 0.322 0.160 0.140 0.000 0.240 0.000 0.121 0.000 0.0 00

FUZZY MULTI-ATTRIBUTE SCORING METHODS WITH APPLICATIONS Cengiz Kahraman1, Semra Birgün2, and Vedat Zeki Yenen2 1 2 Istanbul Technical University, Department of Industrial Engineering, Maçka, stanbu l Istanbul Commerce University, Department of Industrial Engineering, Üsküdar, stanb ul Abstract: The multi-attribute scoring methods are widely used while comparing the alternat ives because of their simplicity. In the case of incomplete information and vagu eness, these multi-attribute scoring methods have been extended to obtain the fu zzy versions. In this chapter, fuzzy simple additive weighting methods and fuzzy multiplicative weighting methods are explained with numerical examples. Scoring , simple additive weighting, multiplicative weighting, multi-attribute Key words: 1. INTRODUCTION An index formulation of a system when the decision maker has a thorough understa nding of the functional relationships among its components, or when he or she po ssesses sufficient data to regress a statistical relationship, can be used in mo deling a multi-attribute problem. Since it often cannot be expected that any of these conditions will be met easily in a normal decision-making environment, thi s chapter presents two scoring techniques: the Simple Additive Weighting method, which obtains an index by adding contributions from each attribute, and the Wei ghted Product method, which obtains the index by multiplying contributions from attributes. 1.1 Crisp Simple Additive Weighting (CSAW) Method The SAW method is probably the best known and most widely used multiple attribut e decision-making (MADM) method. A score in the SAW C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 187

188 C. Kahraman et al. method is obtained by adding contributions from each attribute. Since two items with different measurement units cannot be added, a common numerical scaling sys tem such as normalization is required to permit addition among attribute values. The total score for each alternative then can be computed by multiplying the co mparable rating for each attribute by the importance weight assigned to the attr ibute and then summing these products over all attributes (Yoo and Hwang, 1995). Formally the value of an alternative in the SAW method can be expressed as n V Ai Vi j 1 w j v j xij , i 1,2,..., m (1) where V(Ai) is the value function of alternative Ai and wj and vj (.) are weight and value functions of attribute Xj, respectively. Or the performance of altern ative Ai is calculated by n w j rij V Ai j 1 n (2) wj j 1 where rij is the rating of the ith alternative under the jth attribute with a nu merically comparable scale. Through the normalization process, each incommensura ble attribute becomes a pseudo-value function, which allows direct addition amon g attributes. The value of alternative Ai can be rewritten as Vi n w j rij , i 1, 2,..., m (3) j 1 where rij is the comparable scale of xij, which can be obtained by a normalizati on process. The underlying assumption of the SAW method is that attributes are p referentially independent. Less formally, this means that the contribution of an individual attribute to the total (multiattribute) score is independent of othe r attribute values. Therefore, the decision maker’s preference (or feelings) regar ding the value of one attribute are not influenced in any way by the values of t he other attributes (Fishburn, 1976). Fortunately, studies (Edwards, 1977; Farme

r, 1987) show that the

Fuzzy Multi-Attribute Scoring Methods 189 SAW method yields extremely close approximations to “true” value functions even when independence among attributes does not exactly hold. In addition to the prefere nce independence assumption, the SAW has a required characteristic for weights. That is, the SAW presumes that weights are proportional to the relative value of a unit change in each attribute’s value function (Hobbs, 1980). For instance, let us consider a value function with two attributes: V w1v1 + w2v2. By setting the amount v2/ v1. This of V constant, we can derive the relationship of w1/w2 rela tionship indicates that if w1 0.33 and w2 0.66, the decision maker must be indif ferent to the trade between two units of v1 and one unit of v2. 1.2 Crisp Weighted Product (CWP) Method In the SAW method, addition among attribute values was allowed only after the di fferent measurement units were transformed into a dimensionless scale by a norma lization process. However, this transformation is not necessary if attributes ar e connected by multiplication. When we use multiplication among attribute values , the weights become exponents associated with each attribute value, a positive power for benefit attributes, and a negative power for cost attributes. Formally , the value of alternative Ai is given by (Yoo and Hwang, 1995) n V Ai Vi j 1 x ij j , i w 1, 2 , ..., m (4) Because of the exponent property, this method requires that all ratings be great er than 1. For instance, when an attribute has fractional ratings, all ratings i n that attribute are multiplied by 10m to meet this requirement. Alternative val ues obtained by the multiplicative method do not have a numerical upper bound. T he decision maker may also not find any true meaning in those values. Hence, it may be convenient to compare each alternative value with the standard value. If we use the ideal alternative A* for the comparison purpose, the ratio between an alternative and the ideal alternative is given by n Ri V Ai V A* n xij j j 1 w

x j 1 * j wj , i 1, 2 ,..., m (5)

190 C. Kahraman et al. where xi* is the most favorable value for the jth attribute. It is clear that 0 Ri 1 and the preference of Ai increases when Ri approaches. 2. FUZZY SETS AND FUZZY NUMBERS To deal with vagueness of human thought, Zadeh [1] first introduced the fuzzy se t theory, which was based on the rationality of uncertainty due to imprecision o r vagueness. A major contribution of fuzzy set theory is its capability of repre senting vague knowledge. The theory also allows mathematical operators and progr amming to apply to the fuzzy domain. A fuzzy number is a normal and convex fuzzy set with membership function A ( x ) , which both satisfies normality: A ( x ) =1, for at least one x R , and convexity: A( x ) A ( x1 ) A ( x2 ) where A ( x ) [ 0 ,1 ] and operator. x [ x1 , x2 ] . “ ” stands for the minimization The definition of a triangular fuzzy number has been given in Chapter 4. A flat fuzzy number (FFN) is shown in Figure 1. The membership function ~ of a FFN, V i s defined by ~ ~ ~ ( x V ) ( m1 , f 1 ( y V ) / m2 , m3 / f 2 ( y V ), m4 ) (6) where m1 m2 m3 m4 , f 1 ( y V ) is a continuous monotone increasing ~ ~ function of y for 0 y 1 with f 1( 0 V ) m1 and f 1( 1V ) m2 and ~ f 2 ( y V ) is a conti nuous monotone decreasing function of y for 0 y 1 ~ ~ ~ ( y V ) is denoted simpl y as with f 2 ( 1V ) m3 and f 2 ( 0 V ) m4 . ( m1 / m2 , m3 / m4 ) . y 1.0 ~ f1 ( y V ) ~ f2 ( y V ) ~ 0.0 m1 m2 m3 m4 x Figure 1. A flat fuzzy number, V

Fuzzy Multi-Attribute Scoring Methods 191 3. FUZZY SCORING METHODS Zhang et al. (2001) applied fuzzy logic to compute proximity between an intellec tual property query and a specification, which are tree-structured models constr ucted from their respective Extensible Markup Language representation. Bector et al. (2002) derived a formula for fuzzy scoring model assuming that both the wei ghts and the ratings were fuzzy and demonstrated the use of the main formula wit h a numerical example. Lo (2002) proposed an operating mechanism based on fuzzy theory to integrate fuzzy composite scores of multiple assessments and applied s imulation to test this fuzzy scoring frame. Mitra et al. (2002) described a fuzz y knowledge-based network based on the linguistic rules using the principle of a fuzzy decision tree. They demonstrated the effectiveness of the system on three sets of real-life data. Kwong et al. (2002) introduced a scoring method combine d with a fuzzy expert system in supplier assessment and evaluated their system f or the supplier assessment of electrical appliance products. Belacel and Boulass el (2004) proposed a new multi-criteria fuzzy classification procedure called PR OCFTN, and they also tested this method with an experimental set of 250 cases of astrocytic tumors. Ohdar and Kumar (2004) proposed a fuzzy system to evaluate t he suppliers’ performance. They developed a Genetic Algorithm-based methodology to evolve the optimal set of a fuzzy rule-based system and used a fuzzy inference system of the MATLAB fuzzy logic toolbox to assess the suppliers’ performance. Zha ng (2004) presented the necessity to use fuzzy data for a handover decision in h eterogonous networks and provided new handover criteria along with a new handove r decision strategy. Graf (2005) proposed a game scoring system with the score s ubmission and ranking component, and when compared with statistical approach sho wed that the fuzzy logic approach was more adequate to build into scorings. 3.1 Fuzzy Simple Additive Weighting Methods with Numerical Examples The crisp simple additive weighting method explained above can be transformed to the fuzzy case as follows: When both w j and rij are fuzzy sets wj yj, wj yj , j (7) and

192 rij xij , rij C. Kahraman et al. xij , i, j (8) where y j and xij take their numbers on the real line and w j y j and take value s in [0, 1]. The utility of alternative Ai , rij xij ui ui , u i ui , can be cal culated as follows: The variable u i takes its value on the real line using n and can be obtained y j xij ui j 1 n (9) yj j 1 The membership function n ui ui ui can be calculated using n wj ui sup j 1 yj j 1 rij xij (10) y1 ,..., yn , xi1 ,..., xin where The membership function ui ui is not directly obtainable when w j y j and rij xij are piecewise continuously differentiable fu nctions. To resolve this difficulty and preserve the simplicity of the simple ad ditive weighting method, several approaches have been proposed (Chen et al., 199 2). Some of them are explained in the following discussion. The first four appro aches use the cut to approximate the ui ui . The fifth one, Bonissone’s approach, assumes that all piecewise continuously differentiable fuzzy numbers can be appr oximated by L-R-type trapezoidal numbers. 3.1.1

Baas and Kwakernaak’s Approach It is assumed that w j y j and rij xij are normalized membership functions. The approximate fuzzy utility U i for alternative Ai is determined using the followi ng steps: Step 1. Set an 0 level for ui ui . Step 2. Identify the yj and xij val ues that satisfy

Fuzzy Multi-Attribute Scoring Methods wj 193 0, yj rij xij i, j (11) Step 3. There are many u i values such that ui ui 0 and the extreme ones, u imin and uimax , must be determined. Given a set of real numbers ˆ 1 , , ˆ n , ˆ i1 , , ˆ in such that ˆ and w j ˆ j ˆ ij ui , i , j , y y x x y x ri x ij where ri xij d ri xij dxij (12) and wj yj d wj y j dy j (13) have the same sign. The resulting u i will be either uimax or u imin . A Numeric al Example Assume that we have a decision matrix as follows. X1 X2 A1 excellent fair good A2 good excellent, fair and r21 , r22 good, good . Let th e weight where r11 , r12 important, very important . Figure 2 represents these s et be w1 , w2 linguistic terms. x r22 r12 1.0 0.8 w1 r11 w2 r21

0.5 0.0 x 0.5 0.6 0.7 0.8 0.9 1.0 Figure 2. Fuzzy representation of linguistic terms

194 C. Kahraman et al. 0 Step 1. Let’s set 0.80 .

x x x x y y x Step 2. Let’s identify ˆ 11 , ˆ 12 , ˆ 21 , ˆ 22 , ˆ 1 , ˆ 2 values such that 1 ˆ 11 x x y y 0.80 . The values providing this r12 ˆ 12 r21 ˆ 21 w1 ˆ 1 w2 ˆ 2 equality a re given in the Table 1. Table 1. cut Values While ˆ 21 x 0.72 0.68 0 0.80 ˆ1 y ˆ2 y 0.92 0.88 ˆ 12 x 0.62 0.58 ˆ 11 x 0.82 0.78 ˆ 22 x 0.72 0.68 0.82 0.78 Step 3. There are a total of 24 = 16 possible combinations of ˆ 11 , ˆ 12 , ˆ 1 , ˆ 2 an d ˆ 21 , ˆ 22 , ˆ 1 , ˆ 2 . Using Eq. (9) on all xij and x x y y x x y y y j combination s, we obtain 16 u1 values. The u1 values are given in Table 2. From Table 2, U 1 u1 0.716471 = 0.80 and U 1 u1 0.671765 = 0.80. To calculate u2 values for 0 0.8 0 , the similar operations are made and the following Table 3 is obtained: Table 2. Possible Combinations of xij and yj and Their Corresponding u1 Values x 11 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.82 0.78 0.78 0.78 0.78 0.78 0.78 0.78 0. 78 x12 0.62 0.62 0.62 0.62 0.58 0.58 0.58 0.58 0.62 0.62 0.62 0.62 0.58 0.58 0.5 8 0.58 y1 0.82 0.82 0.78 0.78 0.82 0.82 0.78 0.78 0.82 0.82 0.78 0.78 0.82 0.82 0.78 0.78 y2 0.92 0.88 0.92 0.88 0.92 0.88 0.92 0.88 0.92 0.88 0.92 0.88 0.92 0. 88 0.92 0.88 u1 0.714253 0.716471 0.711765 0.713976 0.693103 0.695765 0.690118 0 .692771 0.695402 0.697176 0.693412 0.695181 0.674253 0.676471 0.671765 0.673976 MAX. MIN.

Fuzzy Multi-Attribute Scoring Methods Table 3. Possible Combinations of xij and yj and Their Corresponding u2 Values x 21 0.72 0.72 0.72 0.72 0.72 0.72 0.72 0.72 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0. 68 x22 0.72 0.72 0.72 0.72 0.68 0.68 0.68 0.68 0.72 0.72 0.72 0.72 0.68 0.68 0.6 8 0.68 0 195 y1 0.82 0.82 0.78 0.78 0.82 0.82 0.78 0.78 0.82 0.82 0.78 0.78 0.82 0.82 0.78 0. 78 y2 0.92 0.88 0.92 0.88 0.92 0.88 0.92 0.88 0.92 0.88 0.92 0.88 0.92 0.88 0.92 0. 88 u2 0.720000 0.720000 0.720000 0.720000 0.698851 0.699294 0.698353 0.698795 0.701 149 0.700706 0.701647 0.701205 0.680000 0.680000 0.680000 0.680000 MAX. MAX. MAX. MAX. MIN. MIN. MIN. MIN. For 0.0; 0.50; and 1.00 values, Tables 4–6 is obtained. Table 4. cut Values While ˆ 21 x ˆ 22 x 0.60 0.80 0 0 .0 0 ˆ 12 x 0.70 0.90 ˆ 11 x 0.50 0.70 ˆ1 y 0.70 0.90 0 ˆ2 y 0.80 1.00 0.60 0.80 Table 5. cut Values While ˆ 22 x 0.65 0.75 0.50 ˆ1 y 0.75 0.85 ˆ 12 x 0.75 0.85

ˆ 11 x 0.55 0.65 ˆ 21 x 0.65 0.75 Table 6. ˆ2 y 0.85 0.95 cut Values While ˆ 22 x 0.70 0.70 0 1.00 ˆ1 y 0.80 0.80 ˆ 12 x 0.80 0.80 ˆ 11 x 0.60 0.60 ˆ 21 x 0.70 0.70 ˆ2 y 0.90 0.90 The final utility results are found for u1 as follows: Table 7. The Utility Values of u1 u1 u1 0 u1max. u1min 0 0.805882 0.582353 0.50 0.744444 0.638235 0.80 0.716471 0.671765 u1 u1 0 0.694118 0.694118

196 C. Kahraman et al. The final utility results are found for u2 as follows: Table 8. The Utility Values of u2 u1 u1 0 u1max. u1min 0 0.80 0.60 0.50 0.75 0.65 0.80 0.72 0.68 u1 u1 0 0.70 0.70 The results in Tables 7 and 8 can be represented in Figure 3: u 1.0 u1 u2 0.5 x Figure 3. The alternatives’ fuzzy utilities using Baas and Kwakernaak’s approach (19 77) The ranking of u1 and u2 can be made by using a proper ranking method. Here, it is clear that u 2 u1 . 3.1.2 Kwakernaak’s Approach (1979) This approach is a modification of Baas and Kwakernaak’s (1977) approach. This app roach proposes an improved algorithm to find the maximum and minimum ui values i nstead of selecting them among all the possible values. For more details, see Ch en et al. (1992). 3.1.3 Dubois and Prade’s Approach (1982) Dubois and Prade (1982) proposed a more efficient search procedure to obtain ui values. This approach assumes that all fuzzy weights wj and fuzzy rij are normal ized fuzzy numbers. The –level sets are used to derive fuzzy

Fuzzy Multi-Attribute Scoring Methods 197 utilities based on the classic SAW method. The steps of this approach are given in the following: Step 1. Set an level and determine – level sets for w j and rij to be wj rij y j , y*j xij , x* ij j i, j (14) (15) Step 2. Compute the normalized fuzzy weights, Pj , j . When the –level sets of w j are known, n – level sets of the normalized fuzzy weights Pj , j , can be obtaine d: P* j y*j y*j k j yk (16) and Pj yj yj k j y* k (17) Let q j p j , p*j n j . Then j 1 qj 1. Step 3. For alternative Ai , the rating rij may be represented by an level set a s in Eq. (15). To order xij and x* , j as ij m1 m2 ... mn – (18) in which m1 min x ij and m n j max x ij and j m1 * in which m1

* m2 * ... mn max x* . ij j * (19) min x* and m* n ij j Step 4. The smallest upper and the largest lower bound of ui are computed as

198 d 1 C. Kahraman et al. p*j m j d 1 uimin j 1 1 j 1 p*j n n p j md j d 1 pj mj j d 1 (20) e 1 uimax j 1 p j m*j e 1 n 1 j 1 pj p*j m* e n p*j m*j (21) j e 1 j e 1 The unknown parameters in Eqs. (20) and (21) are d and e. The parameter d is det ermined when the following equality is satisfied: d 1 n 1

j 1 p* j j d 1 pj zd * pd , pd (22) Similarly, the parameter e is determined when the following equality is satisfie d: e 1 n 1 j 1 pj j e 1 p* j ze * pe , pe (23) Step 5. The fuzzy utility ui can be represented by the interval level. The decis ion maker can set several levels at any and repeat the algorithm several times t o derive an approximated fuzzy utility u i . u imin ,u imax 3.1.4 Cheng and McInnis’s (1980) Approach Cheng and McInnis (1980) pointed out that continuous membership functions of rij and w j are the cause of the complexity of obtaining fuzzy utilities. To avoid such difficulty, they suggested first to convert the continuous membership funct ion to discrete ones and then compute the fuzzy utilities using the following al gorithm: Step 1. The continuous membership function is converted to a discrete o ne. This is done by having the decision maker specify the number of levels that he/she wants to use. The width of intervals is determined according to the decis ion maker’s preference. The decision maker may levels and widths of intervals for different specify different numbers of membership functions in a MCDM problem.

Fuzzy Multi-Attribute Scoring Methods 199 Step 2. For each level, the steps 3 and 4 are performed. The first level to be c onsidered is the largest one among all the w j ’s and rij ’s. level set for each rij and w j can be obtained as: Step 3. Given 0 , the rij and 0 xij , xij* , i, j (24) wj 0 y j , y *j , j (25) Step 4. Given the upper and lower bounds of rij and w j at the 0 –level, the upper and lower bounds of the fuzzy utility at 0 , ui 0 ui min ,ui max , can be compu ted by following the sub-steps: Step 4.1. Compute u i max using the upper bound of rij , j , i.e., x* : ij y j xij ui j yj j (26) To maximize u i , it must be decided whether y j or y*j should be used. Step 4.2 . After finding ui max , u i min can be easily identified. First, xij is used fo r all rij . Second, for those w j whose upper bounds were used for deriving u i max , their lower bounds in computing u i min will be used and vice versa. level until all levels Steps 3 and 4 are used for the next largest are exhausted. 3.1 .5 Bonissone’s (1982) Approach Bonissone (1982) assume that fuzzy/crisp information in decision problems can be approximated by a parameter-based representation. It is called the L-R-type tra pezoidal number (see Figure 4). Fuzzy arithmetic operations with L-R type trapez oidal numbers are given in the following ~ ~ and N c , d , , be positive trapezo idal discussion. Let M a ,b , , fuzzy numbers: ~ M ~ M ~ N ~ N a c, b d , a d , b c,

, , (27) (28)

200 ~ M C. Kahraman et al. 1.0 x a b Figure 4. L-R-Type trapezoidal fuzzy number, M ~ a ,b , , ~ M ~ N ac, bd , a c ,b d . (29) ~ M ~ N a b a d b c , , , d c d d cc (30) Using the algebraic operations above, one can easily compute the performance of an alternative with respect to the attributes by using: n ui j 1 w j rij (31) where wj and rij may be crisp or fuzzy numbers represented in the L-R trapezoida l number format. A Numerical Example Three alternatives of advanced manufacturin g systems, FMS-1, FMS-2, and FMS-3, will be evaluated with respect to four attri butes: engineering effort (X1), flexibility (X2), net present worth (X3), and in tegration ability (X4). The decision matrix is given as X1 FMS1 D = FMS2 FMS3 fair fair very bad X2 good very good very good X3 fair bad very good X4 good good very bad

Fuzzy Multi-Attribute Scoring Methods 201 The weight vector is given as important, more or less important, unimportant, very important ; where very unim portant: (0, 0.2, 0, 0.1); unimportant: (0.3, 0.3, 0.1, 0.1); more or less unimp ortant: (0.4, 0.4, 0.1, 0.1); indifferent: (0.5, 0.5, 0.1, 0.1); more or less im portant: (0.6, 0.6, 0.1, 0.1); important: (0.7, 0.7, 0.1, 0.1); very important: (0.8, 0.8, 0.1, 0.2). The fuzzy set associated with each linguistic term is as f ollows: very bad: (0, 0.2, 0, 0.1); bad: (0.3, 0.3, 0.1, 0.1); more or less bad: (0.4, 0.4, 0.1, 0.1); fair: (0.5, 0.5, 0.1, 0.1); more or less good: (0.6, 0.6, 0.1, 0.1); good: (0.7, 0.7, 0.1, 0.1); very good: (0.8, 0.8, 0.1, 0.2). Then, t he fuzzy utilities for the alternatives are computed as follows: 4 ~ w U1 j 1 w j x1 j (0.7, 0.7, 0.1, 0. 1) 0.7, 0.7, 0.1, 0.1 0.8, 0.8, 0.1, 0.2 0.5, 0.5, 0.1, 0.1 0.3, 0.3, 0.1, 0.1 0.7, 0.7, 0.1, 0.1 0.6, 0.6, 0.1, 0.1 0.5, 0.5, 0.1, 0.1 1.48, 1.48, 0.44, 0.52 4 U2 j 1 w j x2 j (0.7, 0.7, 0.1, 0.1) 0.8, 0.8, 0.1, 0.2 0.8, 0.8, 0.1, 0.2 0.5, 0.5, 0.1, 0.1 0.3, 0.3, 0.1, 0.1 0.7, 0.7, 0.1, 0.1 0.6, 0.6, 0.1, 0.1 0.3, 0.3, 0.1, 0.1 1.48, 1.48, 0.43, 0.66 4 U3 j 1 w j x3 j (0.7, 0.7, 0.1, 0.1) 0.8, 0.8, 0.1, 0.2 0.8, 0.8, 0.1, 0.2 0, 0.2, 0, 0.1 0.3, 0.3, 0.1, 0.1 0, 0.2, 0, 0.1 0.6, 0.6, 0.1, 0.1 0.8, 0.8, 0.1, 0.2 0.72, 1.02, 0.25, 0.62 The obtained fuzzy utilities are illustrated in Figure 5. The ranking of U 1 ,U

2 , and U 3 can be made by using any ranking method. In this example, it is clea r that U 2 is the largest fuzzy number. Thus, FMS2 is selected.

202 x U3 C. Kahraman et al. U2 U3 x 0.47 0.72 1.02 1.05 1.48 1.64 2.00 2.14 Figure 5. The fuzzy utilities of the example problem 3.1.6 Bector et al.’s Approach (2002) Bector et al.’s approach (2002) is a direct treatment with fuzzy numbers to the cr isp case. In this approach, each wi is represented by a triangular fuzzy number given as wi wi 1 , wi 2 , wi 3 , i = 1, 2, . . . , m, whose – cut is given by wi wi1 wi 2 wi1 , wi 3 wi 3 wi 2 (32) and each vij is represented by a triangular fuzzy number given as vij vij 1 ,vij 2 ,vij 3 , j=1,2,…, n, whose – cut is given by m V 1 wi vij (33) m m m Vi i 1

wi1 vij1 , i 1 wi 2 vij 2 , i 1 wi 3 vij 3 . (34)

Fuzzy Multi-Attribute Scoring Methods 203 A Numerical Example (Bector et al., 2002) Discount Saving Bonds (DSB). Bector et al. (2002) assumed that the fuzzy ratings and the fuzzy weights for the DSB are given in the form of TFNs in Table 9 along with their – cuts. Table 9. Computation of Fuzzy Score for Discount Saving Bonds (DSB) Criterion We ight Rating Score Rating Terms & availability Quality of the bonds Backed by sup port =Weight 2 2 2 5 6 5 6 ,4 ,7 ,8 ,7 , 8 4 7 8 , 6 , 9 , 10 8 6 , 24 10 35 12 2 , 63 16 2 48 14 2 , 80 18 2 Liquidity Income frequency from the bonds Trade denominations Taxation Other cha racteristics Recommended for investment Total fuzzy score ( ) 5 4 , 7 , 6 25 10 24 10 2 , 49 14 , 48 14 2

2 2 1 3 2 , 3 , 5 , 4 6 , 8 6 7 2 , 24 11 2 4 4 , 6 , 6 12 7 8 6 2 2 , 30 11 2 2 , 24 10 3 , 5 3 , 5 9 6 2 , 25 10 2 175 78 9 2

, 367 114 9 2 Shown in Table 9, the total fuzzy score ( ) of a DSB is a parabolic fuzzy number . This parabolic fuzzy number can be approximately represented with a triangular fuzzy number by taking 0,1, and 0, respectively: (175, 262, 367).

204 C. Kahraman et al. 3.1.7 Vanegas and Labib’s (2001) Approach Vanegas and Labib (2001) propose a novel method of operating on fuzzy numbers to obtain a fuzzy weighted average of desirability levels during engineering desig n evaluation. The method produces overall desirability levels less imprecise and more realistic than those of the conventional fuzzy weighted average (FWA). The – cut of the overall desirability of an alternative D, calculated through the new FWA for n desirability levels represented by the fuzzy numbers D1 , D2 , , Dn , with weights (fuzzy numbers) W1 ,W2 , ,Wn , is given by D where D a, D b (35) n Di a Da min i 1 n wi (36) wi i 1 and n Di b Db max i 1 n wi (37) wi i 1 where wi Wi a ,Wi b , for all i 1,2 ,..., n and all 0,1 . D a and D b represent the lower and upper limits, respectively, of the – cut Di ; and Di a and Di b repr esent the lower and upper limits, respectively, of the – cut Di ; and Wi a and Wi b represent the lower and upper limits, respectively, of the – cut Wi . The “min” and “m ax” operators take the minimum and maximum values, respectively, that can be calcu lated through the combination of the wi in all the possible ways. The set of wi that is used in the numerator has to be the same as the one in the denominator.

Fuzzy Multi-Attribute Scoring Methods 205 3.2 Fuzzy Multiplicative Weighting Method In the fuzzy case, we can use the fuzzy numbers instead of the crisp ones in Eq. (4). Thus, we have ~ V Ai n w xlijli , j 1 n w x mijmi , j 1 n w xuijui j 1 (38) w where xkijki is the score or value of the kth parameter (k = l, m, and u) of t he criterion j of the alternative i, weighted by the fuzzy weight of the same ~ criterion. The ranking of V A s can be made by using any ranking method. A Numerical Example Two FMS alternatives will be evaluated using the criteria en gineering effort (X1), flexibility (X2), net present worth (X3), and integration ability (X4). The criteria weights and the alternative scores with respect to e ach alternative are given in Table 10. The results of the problem are illustrate d in Figure 6. It is clearly seen that FMS-1 should be selected. Table 10. Data for the Numerical Example Criteria engineering effort (X1) flexib ility (X2) net present worth (X3) integration ability (X4) Criteria Weights (0.1 5, 0.18, 0.24) (0.32, 0.38, 0.46) (0.30, 0.32, 0.38) (0.10, 0.12, 0.18) FMS-1 (3 , 4, 6) (6, 8, 10) (28, 32, 44) (6, 8, 9) FMS-2 (5, 6, 7) (4, 4, 5) (38, 45, 52) (3, 4, 5) V FMS1 30.15 60.32 4 w 4 w w 1 m1 u1 xl1lj , xm1 j , xu1 j j 1 j 1 j 1 280.30 440.38 60.10 , 40.18 80.38 90.18 320.32 80.12 , 4 60.24 100.46 6.8005, 11.0043 , 27.7352

206 C. Kahraman et al. V FMS 2 50.15 7 0.24 40.32 50.46 4 w 4 w w l u2 xl 2 2 , xmm 2 , j j 1 2 j j 1 xu 2 j j 1 380.30 520.38 30.10 , 6 0.18 50.18 40.38 450.32 40.12 , 4 6.5940, 9.3352, 20.0560 What would your selection be if you had used the fuzzy SAW method (Bector et al. , 2002) instead of fuzzy multiplicative weighting method? If you had, you would obtain the following results which indicate that FMS2 should be selected this ti me: ~ V x 1.0 A1 A2 x 6.6 6.8 9.3 11.0 20.0 27.7 Figure 6. Fuzzy multiplicative scores of FMS alternatives ~ V FMS1 ~ V FMS2 11.37 , 14.96 , 24.38 13.73 , 17.48 , 24.64 The reason for this change in the ranking is that the multiplicative model gives a smaller value with x w , 0 w 1 , than the result with x w, 0 w 1 .

Fuzzy Multi-Attribute Scoring Methods 207 4. CONCLUSIONS According to the data type of the alternative’s performance, fuzzy multiattribute decision-making methods can be categorized into three groups: (1) data are all f uzzy, (2) all crisp, and (3) either crisp or fuzzy. The methods in the third gro up are either too cumbersome to use or only suitable for the purpose of screenin g out unsuitable alternatives. The fuzzy MADM methods with data type is all fuzz y require transforming crisp data to fuzzy numbers, despite that the data are cr isp in nature, which not only violates the intention of fuzzy set theory, but al so increases the decision complexity. The fuzzy weighted scoring models are wide ly used in the literature. Simple Additive Weighting Method is probably the best known and widely used method. The overall score of an alternative is computed a s the weighted sum of all the attribute values. It is simple and easy to underst and. Multiplicative weighting methods are superior to SAW methods because they d o not need the data to be normalized. The data with different units can be direc tly used in multiplicative methods. REFERENCES Baas, S.M., and Kwakernaak, H., 1977, Rating and ranking of multiple aspect alte rnative using fuzzy sets, Automatica, 13: 47 58. Bector, C.R., Appadoo, S.S., an d Chandra, S., 2002, Weighted factors scoring model under fuzzy data, Proceeding of the Annual Conference of the Administrative Sciences Association of Canada M anagement Science Division, ed. Kumar, U., 95 105, Winnipeg, Manitoba. Belacel, N., and Boulassel, M.R., 2004, Multicriteria fuzzy classification procedure proc ftn: methodology and medical application, Fuzzy Sets and Systems, 141(2): 203 21 7. Bonissone, P.P., 1982, A fuzzy set based linguistic approach: Theory and appl ications, in Approximate Reasoning in Decision Analysis, Gupta, M.M., and Sanche z, E., eds., pp: 329 339, Elsevier. Chang, Y.M., and McInnis, B., 1980, An algor ithm for multiple attribute, multiple alternative decision problem based on fuzz y sets with application to medical diagnosis, IEEE Transactions on System, Man, and Cybernetics, SMC-10: 645 650. Chen, S-J., Hwang, C-L., and Hwang, F.P., 1992 , Fuzzy Multiple Attribute DecisionMaking: Methods And Applications, Springer Ve rlag, Heidelberg. Dubois, D., and Prade, H., 1982, The use of fuzzy numbers in d ecision analysis, in Fuzzy Information and Decision Processes, Gupta, M.M., and Sanchez, E., eds., 309 321, North-Holland, Amsterdam.

208 C. Kahraman et al. Edwards, W., 1977, How to use multiattribute utility measurement for social deci sion making, IEEE Transactions on Systems, Man and Cybernetics, SMC-7: 326–340. Fa rmer, T.A., 1987, Testing the robustness of multi-attribute utility theory in an applied setting, Decision Sciences, 18(2): 178–193. Fishburn, P.C., 1976, Noncomp ensatory preferences. Synthese, 33: 393–403. Graf, A., 2005, Fuzzy logic approach for modelling multiplayer game scoring system, 8th International Conference on T elecommunications ConTEL 2005, pp: 347 352, Zagreb, Croatia. Hobbs, B.F., 1980, A comparison of weighting methods in power plant siting, Decision Sciences, 11: 725–737. Kwong, C.K., Ip, W.H., and Chan, J.W.K., 2002, Combining scoring method a nd fuzzy expert systems approach to supplier assessment: a case study, Integrate d Manufacturing Systems, 13(7): 512 519. Kwakernaak, H., 1979, An algorithm for rating multiple-aspect alternatives using fuzzy sets, Automatica, 15: 615 616. L o, H.C., 2002, A preliminary study of development of fuzzy composite score for m ultiple assessments, Chinese Journal of Science Education, 10(4): 407 421. Mitra , S., Konwar, K.M., and Pal, S.K., 2002, Fuzzy Decision Three, Linguistic Rules and Fuzzy Knowledge-Based Network: Generation and Evaluation, IEEE Transactions on Systems, Man, and Cybernetic Part C: Applications and Reviews, 32(4): 328 339 . Ohdar, R., and Kumar, P.R., 2004, Performance measurement and evaluation of su ppliers in supply chain: an evolutionary fuzzy-based approach, Journal of Manufa cturing Technology, Management, 15(8): 723 734. Vanegas, L.V., and Labib, A.W., 2001, Application of new fuzzy-weighted average (NFWA) method to engineering des ign evaluation, International Journal of Production Research, 36(6): 1147 1162. Yoo, K.P., and Hwang, C-L., 1995, Multiple Attribute Decision-Making: An Introdu ction, Sage University Publications, California. Zhang, T., Benini, L., and De M icheli, G., 2001, Component selection and matching for ipbased design, Proceedin gs of the Design and Test in Europe, March: 40 46. Zhang, W., 2004, Handover Dec ision Using Fuzzy MADM in Heterogeneous Networks, WCNC 2004, IEEE Communications Society, 653–658. Zadeh, L., 1965, Fuzzy sets, Information Control, 8: 338–353.

FUZZY MULTI-ATTRIBUTE DECISION MAKING USING AN INFORMATION AXIOM-BASED APPROACH Cengiz Kahraman1 and Osman Kulak2 1 2 Istanbul Technical University, Industrial Engineering Department, Macka, Istanbu l, Turkey Pamukkale University, Industrial Engineering Department, Denizli, Turk ey Abstract: Axiomatic design (AD) provides a framework to describe design objects and a set of axioms to evaluate relations between intended functions and the means by whic h they are achieved. Since AD has the characteristics of multi-attribute evaluat ion, it is proposed for multi-attribute comparison of some alternatives. The com parison of these alternatives is made for the cases of both complete and incompl ete information. The crisp AD approach for complete information and the fuzzy AD approach for incomplete information are developed. In this chapter, the numeric applications of both crisp and fuzzy AD approaches for the comparison of flexib lemanufacturing systems are given. Axiomatic design, multi-attribute, informatio n axiom, flexible manufacturing Key words: 1. INTRODUCTION Approaches that include more than one measure of performance in the evaluation p rocess are termed multi-attribute or multi-criteria decision methods. The advant age of these methods is that they can account for both financial and nonfinancia l impacts. Among these methods, the most popular ones are scoring models (Nelson , 1986), analytic hierarchy process (AHP) (Kahraman et al., 2004), analytic netw ork process (ANP) (Büyüközkan et al., 2004), utility models (Suh, 1995), order prefere nce by similarity ideal solution (TOPSIS) (Deng et al., 2000), and outranking me thods (De Boer et al., 1998). Axiomatic design principles, including the C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 209

210 C. Kahraman and O. Kulak information axiom also, present an opportunity for multi-attribute evaluation. T he axiomatic design process is described by the mapping process from functional requirements (FRs) to design parameters (DPs). The goal in axiomatic design is t o satisfy the goals of the customer domain through accomplishment in the subsequ ent domains, which requires mapping from one space to the next. In the mapping ( design) process, Suh (1990) imposes two axioms that must be followed in order to create the “best” design. The information axiom (IA), which is the second axiom of AD, proposes the selection of the proper alternative that has minimum informatio n content. Having to use crisp values is one of the problematic points in the cr isp evaluation process. As some criteria are difficult to measure by crisp value s, they are usually neglected during the evaluation. Another reason is about mat hematical models that are based on crisp values. These methods cannot deal with decision makers’ ambiguities, uncertainties, and vagueness, which cannot be handle d by crisp values. The use of fuzzy set theory (Zadeh, 1965) allows the decision makers to incorporate unquantifiable information, incomplete information, nonob tainable information, and partially ignorant facts into the decision model. A mo del based on IA enables decision makers to evaluate both qualitative and quantit ative criteria together. In this chapter, a crisp multiattribute information axi om (IA) approach and then a fuzzy multi-attribute IA approach for multi-attribut e decision-making problems will be developed and the implementation process will be shown by the real-world examples. 2. PRINCIPLES OF AXIOMATIC DESIGN The most important concept in axiomatic design is the existence of the design ax ioms. The first design axiom is known as the independence axiom, and the second axiom is known as the information axiom. They are stated as follows (Suh, 1990). Axiom 1. The Independence Axiom: Maintain the independence of functional requir ements. Axiom 2. The Information Axiom: Minimize the information content. The in dependence axiom states that the independence of FRs must always be maintained w here FRs are defined as the minimum set of

Fuzzy MADM Using Information Axiom 211 independent requirements that characterize the design goals. The information axi om states that the design with the smallest information content among those sati sfying the first axiom is the best design (Suh, 2001). 2.1 Crisp Information Axiom Information is defined in terms of the information content, I, that is related i n its simplest form to the probability of satisfying the given FRs. Information content I i for a given FR i is defined as follows: I i = log 2 1 pi (1) where p i is the probability of achieving the functional requirement FR i and lo g is the logarithm in base 2 (with the unit of bits). This definition of informa tion follows the definition of Shannon (1948), although there are operational di fferences. Because there are n FRs, the total information content is the sum of all these probabilities. If Ii approaches infinity, the system will never work. When all probabilities are one, the information content is zero, and conversely, the information required is infinite when one or more probabilities are equal t o zero (Suh, 1995). In any design situation, the probability of success is given by what the designer wishes to achieve in terms of tolerance (i.e., design rang e) and what the system is capable of delivering (i.e., system range). As shown i n Figure 1, the overlap between the designer-specified “design range” and the system capability range “system range” is the region where the acceptable solution exists. Therefore, in the case of a uniform probability distribution function, p i may be written as pi = Common range . System range (2) Therefore, the information content is equal to I i = log 2 System range . Common range (3)

212 C. Kahraman and O. Kulak The probability of achieving FR i in the design range may be expressed, if FR i is a continuous random variable, as pi = 2 1.8 1.6 Probability density dr dr u p s ( FR ).dFR 1 (4) 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 5 10 Design Range System pdf Common Range Common Area 15 20 FR 25 30 35 40 45 Figure 1. Design range, system range, common range, and probability density func tion (pdf) of an FR where ps (FR) is the system pdf (probability density function) for FR. Eq. (4) g ives the probability of success by integrating the system pdf over the entire de sign range. (i.e., the lower bound of design range, dr 1 , to the upper bound of the design range, dr u ). In Figure 2, the area of the common range ( Acr ) is equal to the probability of success P (Suh, 1990). Therefore, the information co ntent is equal to I log 2 1 . Acr (5) The information content in Eq. (1) is a kind of entropy that measures uncertaint y. There are some other measures of information in terms of uncertainty. Prior t o the theory of fuzzy sets, two principal measures of uncertainty were recognize d. One of them, proposed by Hartley (1928), is based solely on the classic set t heory. The other, introduced by Shannon (1948), is formulated in terms of probab ility theory. Both of these measures pertain to some aspects of ambiguity, as op posed to vagueness or fuzziness. Both Hartley and Shannon introduced their measu res for the

Fuzzy MADM Using Information Axiom 213 purpose of measuring information in terms of uncertainty. Therefore, these measu res are often referred to as measures of information. The measure invented by Sh annon is referred to as the Shannon entropy. The Shannon entropy, which is a mea sure of uncertainty and information formulated in terms of probability theory, i s expressed by the function Probability 0.4 Density 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 Area with in common range (Acr) Design Range System pdf 3 2 Common Range 1 0 System Range 1 2 FR 3 Figure 2. Design range, system range, common range, and pdf of a FR H p x /x X x X p x log 2 p x (6) where p x / x X is a probability distribution on a finite set X. Suh’s entropy in axiomatic design does not require that the total of the probabilities be equal t o 1.0, whereas Shannon entropy does. Because of this property, Shannon entropy s hould not be used as an entropy measure while evaluating independent functional requirements in axiomatic design. 2.2 Fuzzy Information Axiom Approach The multi-attribute crisp information axiom approach mentioned before can be use d for the solution of decision-making problems under certainty. This approach ca nnot be used with incomplete information, since the expression of decision varia bles by crisp numbers would be ill defined. For this reason, the multi-attribute fuzzy information axiom is developed in this study. At the same time, a problem including both crisp and fuzzy criteria can be solved by integrating crisp and fuzzy information axiom

214 C. Kahraman and O. Kulak approaches. This feature is an important advantage that can not be found in othe r multi-attribute approaches. The definition and formulation of the developed fu zzy approach are given in the following discussion. (x) Figure 3. The numerical approximation system for intangible factors The data relevant to the criteria under incomplete information can be expressed as fuzzy data. The fuzzy data can be linguistic terms, fuzzy sets, or fuzzy numb ers. If the fuzzy data are linguistic terms, they are transformed into fuzzy num bers first. Then all the fuzzy numbers (or fuzzy sets) are assigned crisp scores . The following numerical approximation systems are proposed to systematically c onvert linguistic terms into their corresponding fuzzy numbers. The system conta ins five conversion scales (Figures 3 and 4). (x) Very Low Low Medium High Very High 1 1 1 1 1 1 1 1 1 1 X Figure 4. The numerical approximation system for tangible factors In the fuzzy case, we have incomplete information about the system and design ra nges. The system and design range for a certain criterion will

Fuzzy MADM Using Information Axiom 215

be expressed by using “over a number,” “around a number,” or “between two numbers.” Triangu ar or trapezoidal fuzzy numbers can represent these kinds of expressions. We now have a membership function of a triangular or trapezoidal fuzzy number, whereas we have a probability density function in the crisp case. So, the common area i s the intersection area of triangular or trapezoidal fuzzy numbers. The common a rea between design range and system range is shown in Figure 5. (x) TFN of Systen Design Common Area TFN of Design Range X Figure 5. The common area of system and design ranges Therefore, information content is equal to I log 2 TFN of System Design . Common Area (7) In the following section, the numerical application of these approaches for solv ing multi-attribute decision-making problems is given. 3. MULTI-ATTRIBUTE COMPARISON OF ADVANCED MANUFACTURING SYSTEMS The term “advanced manufacturing systems” (AMS) is broadly defined to include any au tomated (usually computer oriented) technology used in design, manufacturing/ser vice, and decision support. Components of AMS include computer-aided engineering , factory management and control

216 C. Kahraman and O. Kulak systems, computer-integrated manufacturing processes, and information integratio n. Many factories reach an intermediate stage, often-called flexible manufacturi ng systems (FMS). At this stage some machine tools, material-handling equipment, and other programmable devices are under the integrated control of a computer. FMS can manufacture a wide range of products in batch sizes from one to thousand s. They provide many important benefits such as greater manufacturing flexibilit y, reduced inventory, reduced floor space, faster response to shifts in market d emand, lower lead times, and a longer useful life of equipment over successive g enerations of products. Like many real-world problems, the decision of investing in advanced manufacturing technology frequently involves multiple and conflicti ng objectives, e.g., minimizing costs, maximizing flexibility, minimizing machin e downtimes, or maximizing efficiency (Kulak and Kahraman, 2005). 3.1 A Numerical Application of Crisp Information Axiom A company manufacturing tractor components wants to renew the manufacturing syst em. In order to produce a group of products, the company must decide to select t he most appropriate one among the different alternative flexible manufacturing s ystems. With respect to the characteristics of the product group manufactured by a company, the functional requirements that should be satisfied by a flexible m anufacturing system are given below. Since FR1 is a monetary criterion, it is a different criterion from the others. The other criteria are graded between 1 and 20. This grading is arranged to show that the interval 17 20 is excellent, 13 1 6 is very good, 9 12 is good, 5 8 is fair, and 1 4 is poor. FR1= Annual Deprecia tion and Maintenance Cost (ADMC) must be in the range of $100,000 to $200,000, F R2= Quality of Results (QR) must be over 9, FR3= Ease of use (EU) must be over 1 3, FR4= Competitive (C) must be in the range of 15 to 18, FR5= Adaptability (A) must be over 15, FR6= Expandability (E) must be in the range of 12 to 16. Altern ative flexible-manufacturing systems’ annual depreciation and maintenance costs an d performance scores evaluated by the experts with

Fuzzy MADM Using Information Axiom 217 respect to certain criteria are given in Table 1. The data given in the Table 1 are arranged to include the minimum and maximum performance values supplied by t he system. Table 1. The System Range Data for Advanced Manufacturing Systems AMSs FMS-I FMS-II FMS-III FMS-IV ADMC (*$1000) 210 to 240 80 to 120 180 to 220 1 40 to 170 QR 18 to 20 12 to 17 8 to 12 7 to 10 EU 13 to 18 9 to 14 10 to 14 8 to 14 C 16 to 20 12 to 17 13 to 18 13 to 17 A 12 to 18 15 to 17 19 to 20 12 to 16 E 12 to 16 14 to 18 9 to 14 11 to 13 The data in Table 1, related to annual depreciation and maintenance costs, refle ct only the minimum and maximum cost values. The ADMC costs of the alternatives in Table 1 have the probability density functions as shown in Table 2. Table 2. The Probability Density Functions of ADMC AMS FMS-I FMS-II FMS-III FMSIV The Probability Density Functions of ADMC Range ($100,000) f f f f x x x x 0.697 x 2 2.404 x 3 0.723x 2 1.591x 2 0.2 0.4 0.5 2.1 x 2.4 0.8 x 1.2 1.8 x 2.2 1.4 x 1.7 Using these design and system ranges, the information content for each FR in eac h FMS may be computed using Equations (3) and (5). Some sample calculations to o btain the information contents of ADMC and QR are presented below. Annual Deprec iation and Maintenance Cost For FMS-I: Acr = 0 I ADMC 1 log 2 1 = Acr (8)

218 C. Kahraman and O. Kulak For FMS-II: 1.2 Acr 1 2.404x 3 dx 0.645 I ADMC 2 log 2 1 = 0.633 A cr (9) The design and system ranges of ADMC for FMS-I and FMS-II are shown in Figures 6 and 7, respectively. And the design and systems ranges of QR for FMS are also s hown in Figures 8–13. 4 3.5 3 2.5 f(x) 2 1.5 1 0.5 0 1 1.2 1.4 1.6 1.8 2 2.2 x ($100000) 2.4 2.6 2.8 3 Design Range System Range f1(x) Figure 6. Design and system ranges of ADMC for FMS-I For FMS-III: 2 Acr 1.8 0.723 x 2 2 dx 5 0.442 I ADMC 3 log 2 1 =1.178 (10) Acr For FMS-IV: 1.7 Acr 1.4 1.591x 2 1 dx 1 I ADMC 2 4

log 2 1 = 0 (11) Acr

Fuzzy MADM Using Information Axiom 4.5 4 3.5 3 f(x) 219 Design Range System Range 2.5 2 1.5 1 0.5 0 0.8 f2(x) Common Area 1 1.2 1.4 x ($100000) 1.6 1.8 2 Figure 7. Design and system ranges of ADMC for FMS-II Quality of Results For FMS-I: I QR 1 log 2 20 18 20 18 log 2 1 0 (12) 3 2.5 2 f(x) System Range Design Range f3(x) 1.5 1 0.5 0 1 1.2 1.4 Common Area 1.6 x ($100000) 1.8 2 2.2

Figure 8. Design and system ranges of ADMC for FMS-III

220 4.5 4 3.5 3 f(x) Design Range System Range f4(x) C. Kahraman and O. Kulak 2.5 2 1.5 1 0.5 0 0.8 1 1.2 1.4 1.6 x ($100000) 1.8 2 Common Area Figure 9. Design and system ranges of ADMC for FMS-IV For FMS-II: I QR 2 log 2 17 12 17 12 log 2 1 0 (13) Figure 10. Design and system ranges of QR for FMS-I

Fuzzy MADM Using Information Axiom 221 Figure 11. Design and system ranges of QR for FMS-II For FMS-III: I QR 3 log 2 12 8 12 9 log 2 4 3 0, 415 (14) Figure 12. Design and system ranges of QR for FMS-III For FMS-IV: I QR 4 log 2 10 7 10 9 log 2 3 1,585 (15)

222 C. Kahraman and O. Kulak Figure 13. Design and system ranges of QR for FMS-IV The information contents for the other criteria with respect to the alternatives are given in Table 3. As the system with minimum information content is the bes t one, FMS-II is selected. Table 3. Suh’s Information Content for Advanced Manufacturing Systems AMSs FMS-I F MS-II FMS-III FMS-IV IADMC Infinite 0.633 1.178 0.000 IQR 0.000 0.000 0.415 1.58 5 IEU 0.000 2.322 2.000 2.585 IC 0.415 1.322 0.737 1.000 IA 1.000 0.000 0.000 2. 000 IE 0.000 1.000 1.322 1.000 Ii Infinite 5.277* 5.652 8.170 3.2 A Numerical Application of Fuzzy Information Axiom The same company in Section 3.1 has the following fuzzy functional requirements: FR1 = ADMC must be low, FR2 = QR must be very good, FR3 = EU must be very good, FR4 = C must be excellent, FR5 = A must be excellent, FR6 = E must be very good .

Fuzzy MADM Using Information Axiom 223 The experts produce the system range data and use the linguistic expressions as in Table 4. Table 4. The System Range Data for Advanced Manufacturing Systems AMS ADMC QR EU C A E FMS-I FMS-II FMS-III FMS-IV High Very Low Medium Low Excellent Very good Good Fair Very good Good Good Good Excellent Very good Very good Very good Very good Very good Excellent Very good Very good Very good Good Good The conversation scales for intangibles are given in Figure 14 whereas the ones for ADMC are given in Figure 15. In order to obtain the information content for ADMC and QR two sample calculations are given in the following. Annual Depreciat ion and Maintenance Cost For FMS-III: Common Area = (180 170) × 0.2 / 2 = 1 System Area = (210 170) × 1 / 2 = 20 I ADMC log 2 System Area Common Area Log 2 20 1 4.322 (16) Figure 14. TFNs for intangible factors

224 C. Kahraman and O. Kulak Figure 15. TFNs for tangible factors Quality of results For FMS-III: Common Area = (14 12) × 0.333 / 2 = 0.333 System A rea = (14 8) × 1 / 2 = 3 I QR log 2 System Area Common Area log 2 3 0.333 3.171 (17) Figure 16. Design and system ranges of ADMC in case of fuzziness

Fuzzy MADM Using Information Axiom 225 Figure 17. Design and system ranges of QR in case of fuzziness The information contents for the other criteria with respect to the alternatives are given in Table 5. The alternative with minimum information content is FMS-I I. Table 5. The information content for advanced manufacturing systems AMSs FMS-I F MS-II FMS-III FMS-IV IADMC Infinite 5.322 4.322 0.000 IQR 2.806 0.000 3.171 Infi nite IEU 0.000 3.171 3.171 3.171 IC 0.000 3.391 3.391 3.391 IA 3.391 3.391 0.000 3.391 IE 0.000 0.001 3.171 3.171 Ii Infinite 15.275* 17.000 Infinite The rankings obtained by using the crisp and fuzzy approaches are the same. When the attribute ADMC is excluded in the evaluation above, FMS-I will be the best alternative. Although FMS-I is the best alternative having the minimum informati on content in total for all the criteria except ADMC, FMS-I is not selected sinc e the ADMC system and design ranges are not overlapped. 4. MULTI-ATTRIBUTE EQUIPMENT SELECTION The satisfaction of customer requirements forces companies to become more sensit ive and to make deep analyses in selecting equipment. The selection of oversized equipment can disturb the company’s cash flow and also the problems such as exces sive inventory and idle equipment can be met. On the contrary, the selection of under-sizing equipment cannot fulfill requested quality levels and capacity requ irements by customers.

226 C. Kahraman and O. Kulak Equipment selection is also an important decision-making problem for the design of a flexible manufacturing system (Kulak et al., 2005) An international company needs a few punching machines to manufacture its products, racks, and sub-racks in which electronic materials are located. The company determined six possible punching machines with respect to the manufacturing requirements. The criteria c onsidered in the selection process are categorized into the groups of costs and technical characteristics. The group of costs includes fixed costs per hour, var iable costs per hour, and equivalent costs of standard tools per hour. The group of technical characteristics includes length of sheet size, thickness of sheet metal, number of strokes for 25-mm pitchsize sheet metal, simultaneous axis spee d, tool rotation speed, and sufficiency of service. Some criteria including posi tioning the work piece precisely and width of sheet metal are excluded since the values of these criteria are the same for each candidate. The criteria in the g roup of costs are linguistic variables. The sufficiency of service in the group of technical characteristics is also a linguistic variable. The company’s design r anges, which means that what a designer wants to achieve for the above criteria are as follows: FRFC = Fixed costs per hour (FC) must be medium, FRVC = Variable costs per hour (VC) must be low, FRST = Equivalent costs of standard tools per hour (ST) must be low, FRL = Length of sheet size (L) must be in the range of 12 00 to 2540, FRT = Thickness of sheet metal (T) must be in the range of 3 to 8, F RNS = Number of strokes for 25 mm pitchsize sheet metal (NS) must be in the rang e of 190 to 445, FRXY = Simultaneous axis speed (XY) must be in the range of 70 to 110, FRSR = Tool rotation speed (SR) must be in the range of 50 to 180, FRSS = Sufficiency of service (SS) must be excellent. Alternative punching machines’ co sts and performance scores evaluated by the company’s managers with respect to cri teria are given in Tables 6 and 7. The data for design ranges and the data for s ystem ranges are entered into the software-MAXD. The calculated results below ar e obtained by MAXD. The data given in Table 7, except sufficiency of service, ar e arranged to include the minimum and maximum performance values supplied by the punching machines. The managers produce the system range data and use the lingu istic expressions about costs and sufficiency of service as in Tables 6 and 7.

Fuzzy MADM Using Information Axiom 227 Figures 18, 19, and 20 show the membership functions of the linguistic expressio ns about fixed costs per hour, variable costs per hour, and equivalent costs of standard tools per hour, respectively. Figure 21 also shows the membership funct ions of the linguistic expressions about sufficiency of service. For example, in Figure 18, the decision maker subjectively evaluates the alternatives with the linguistic term “very low” if it is assigned a score of (8, 8, 10); “low” with a score o f (8, 10, 12); “medium” with a score of (10, 12, 14); “high” with a score of (12, 14, 16 ); and “very high” with a score of (14, 16, 16). Table 6. The System Range Data for Costs Alternative Punch Equipments Criteria V ariable costs per hour (Euro/hour) Very Low Medium Medium High Medium Low Fixed costs per hour (Euro/hour) Low Medium High Very high Medium Low Equivalent costs of standard tools per hour (Euro/hour) Low Low Low Low High Med ium Punch-A Punch-B Punch-C Punch-D Punch-E Punch-F Table 7. The System Range Data for Technical Characteristics Alternative Criteri a Punching Length of Thickness Number of Simultaneous strokes for axis speed Mac hines sheet size of sheet (m/min.) metal (mm) 25 mm (mm) pitchsize sheet metal P unch-A 0 to 1270 0 to 6,4 0 to 420 0 to 108 Punch-B 0 to 2070 0 to 6,4 0 to 220 0 to 97 Punch-C 0 to 2540 0 to 6,4 0 to 445 0 to 108 Punch-D 0 to 2535 0 to 8,0 0 to 445 0 to 108 Punch-E 0 to 2500 0 to 6,4 0 to 400 0 to 110 Punch-F 0 to 1270 0 to 6,4 0 to 200 0 to 82 Sufficiency Tool of service rotation speed (rpm) 0 to 180 0 to 60 0 to 180 0 to 60 0 to 60 0 to 60 Excellent Excellent Excellent Excellent Very Good Very Good

228 C. Kahraman and O. Kulak Figure 18. TFNs for tangible factors (fixed costs) Figure 19. TFNs for tangible factors (variable costs) Figure 20. TFNs for tangible factors (equivalent costs of standard tools per hou r)

Fuzzy MADM Using Information Axiom 229 Figure 21. TFNs for intangible factors (sufficiency of service) In the following section, unweighted and weighted multi-attribute IA approaches will be applied to the equipment selection problem above. 4.1 Unweighted Multi-Attribute IA Approach The information content for Punch-B can be computed using Eq. (3) with the syste m range in Table 7 and the design range for the length of metalsheet (L) above. For Punch-B : Common Area = (2070 1200) × 1 = 870 System Area= (2070 0) × 1 = 2070 IL log 2 System Area Common Area log 2 2070 870 1.250 (18) The information content for Punch-A can be computed using Eq. (7) with the syste m range in Table 6 and the design range for the fixed costs (FC). For Punch-A : Common Area= (12 10) × 0.5 / 2 = 0.5 System Area= (12 8) × 1 / 2 = 2 I FC log 2 System Area Common Area log 2 2 0.5 2.000 (19)

230 C. Kahraman and O. Kulak For this approach, the results in Table 8 are obtained in a similar way that the sample numerical results are calculated. Table 8. The Results of Suh’s Information Content for Punching Machines Punching IFC Machines A B C D E F 2.000 0.000 2.000 Infinite 0.000 2.000 IVC 1.0 00 2.000 2.000 2.000 0.000 IST 0.000 0.000 0.000 Infinite 2.000 IL 4.181 1.250 0 .923 0.925 0.943 4.181 IT 0.912 0.912 0.912 0.678 0.912 0.912 INS 0.869 2.874 0. 803 0.803 0.930 4.322 IXY 1.507 1.845 1.507 1.507 1.459 2.773 ISR 0.470 2.585 0. 470 2.585 2.585 2.585 ISS 0.000 0.000 0.000 0.000 3.391 3.391 I 10.939 11.467 8.615* Infinite Infinite 22.164 Infinite 0.000 The information contents for the criteria with respect to the alternatives are g iven in Table 8. As the punching machine with minimum information content is the most suitable alternative with respect to the designer’s requirements, Punch-C is selected. In Table 9, the information contents for costs and technical characte ristics are given separately since a decision maker may require seeing the effec t of any main criterion (costs or technical characteristics). Punch-B is the mos t suitable alternative with respect to the designer’s requirements when the main c riteria costs are only taken into account. Punch-C is the most suitable alternat ive with respect to the designer’s requirements when the main criteria technical c haracteristics are only taken into account. Table 9. The Results of Suh’s Information Content For Costs And Technical Characte ristics Punch.Mach. IFC A B C D E F 2.000 0.000 2.000 0.000 2.000 IVC 1.000 2.000 2.000 2.000 0.000 IST 0.000 0.000 0.000 Costs IL 3.000 2.000* 4.000 IT Technical Chara cteristics I INS 0.869 2.874 0.803 0.803 0.930 4.322 IXY 1.507 1.845 1.507 1.507 1.459 2.773 ISR 0.470 2.585 0.470 2.585 2.585 2.585 ISS 0.000 0.000 0.000 0.000 3.391 3.391 I 7.939 9.467 4.615* 6.498 10.221 18.164 4.181 0.912 1.250 0.912 0.923 0.912 Infinite Infinite 0.000 2.000 Infinite 0.925 0.678 4.000 4.181 0.912 Infinite Infinite 0.943 0.912 Since each main criterion involves different numbers of subcriteria, the effect of each main criterion on the sum of information contents in Table 8 will possib ly be different. In order to remove this effect, the decision maker may use unit indexes for unweighted information content given in Table 10, which are calcula ted by dividing the total information contents

Fuzzy MADM Using Information Axiom 231 in Table 9 by the number of subcriteria of each main criterion. The column of to tal unit index in Table 10 is calculated by summing the unit indexes for costs a nd technical characteristics. Table 10. Unit Indexes for Unweighted Information Contents Punching Machines A B C D E F Index for Costs 1.000 0.667 1.333 Infinite Infinit e 1.333 Index for Tech. Characteristics 1.323 1.578 0.769 1.083 1.703 3.027 Tota l Unit Index 2.323 2.245 2.103* Infinite Infinite 4.361 With respect to the total unit indexes, Punch-C is the selected alternative. Alt hough the same alternative is selected in Table 8 and in Table 10, different alt ernatives might have been selected. The effect of this approach will be seen in the following section when the weighted multiattribute IA approach is used. 4.2 Weighted Multi-Attribute IA Approach In the method in subsection 4.1., the weights for all subcriteria are equal. If the decision maker wants to assign a different weight for each criterion, the fo llowing weighted multi-attribute IA approach can be used. Eq. 20 is proposed for the weighted multi-attribute IA approach: 1 p ij 1 p ij 1 w j w j , , I ij I ij 1 1 log 2 I ij , 0 I ij 1 (20) log 2 w j For this approach, the results in Table 11 are obtained by e data in Table 8. The weights for the main criteria costs ments are determined as 0.80 and 0.20, respectively, since company give higher importance to the product prices than acteristics. applying Eq. 20 to th and technical require the customers of this to the technical char

232 C. Kahraman and O. Kulak Table 11. The Weighted Results for Cost and Technical Characteristics Costs IL 2.741 1.741 3.482 Infinite Infinite 3.482 Punch IFC Mach. A B C D E F 1.741 0.000 1.741 Infinite 0.000 1.741 IVC 1.000 1.741 1.741 Infinite 1.741 0.000 IST 0.000 0.000 0.000 0.000 IT 1.331 1.046 0.670 0.678 0.747 1.331 I 0.632 0.632 0.632 0.143 0.632 0.632 INS 0.495 1.235 0.334 0.334 0.695 1.340 Technical Characteristics IXY ISR ISS 1.085 1.130 1.085 1.085 1.079 1.226 0.023 1.209 0.023 1.209 1.209 1.209 0.000 0. 000 0.000 0.000 1.277 1.277 I 3.566 5.252 2.744 3.450 5.638 7.015 Infinite 1.741 Punch-B is selected when unit indexes for weighted information contents in Table 11 are used. Table 12 gives unit indexes for weighted information contents. The ranking order when the unweighted approach is used changes as in Table 12 in fa vor of Punch-B, since the weighted approach reflects the high importance of cost s to the results. Table 12. Unit Indexes for Weighted Information Contents Punching Machines A B C D E F Index for Costs 0.914 0.580 1.161 Infinite Infinit e 1.161 Index for Tech. Characteristics 0.594 0.875 0.457 0.575 0.940 1.169 Tota l Unit Index 1.508 1.456* 1.618 Infinite Infinite 2.330 5. CONCLUSION Crisp multi-attribute decision making (MADM) methods solve problems in which all decision data are assumed to be known and must be represented by crisp numbers. The methods are to effectively aggregate performance scores. Fuzzy MADM methods have difficulty in judging the preferred alternatives because all aggregated sc ores are fuzzy data. We propose a crisp multi-attribute IA approach when all dec ision data are known, whereas we propose fuzzy multi-attribute IA approach when unquantifiable or incomplete information exists. The proposed crisp and fuzzy IA approaches use the design ranges determined by the decision makers to select th e best alternative. However, these approaches that depend on the minimum informa tion axiom do not let an alternative be selected even if that alternative meets the design ranges of all other criteria

Fuzzy MADM Using Information Axiom 233 successfully but not any of these ranges. However, the decision maker can assign a numerical value instead of an “infinitive” in order to make the selection of an a lternative possible which meets all other criteria successfully, except the crit erion having an “infinitive” value. REFERENCES Büyüközkan, G., Ertay, T., Kahraman, C., and Ruan, D., 2004, Determining the importanc e weights for the design requirements in the house of quality using the fuzzy an alytic network approach, International Journal of Intelligent Systems, 19(5): 44 3–461. De Boer, L., Van Der Wegen, L., and Jan Telgen, J., 1998, Outranking method s in support of supplier selection, European Journal of Purchasing and Supply Ma nagement, 4(2–3): 109–118. Deng, H., Yeh, C.H., and Willis, R.J., 2000, Inter-compan y comparison using modified TOPSIS with objective weights, Computers & Operation s Research, 27: 963–973. Hartley, R.V.L., 1928, Transmission of information, The B ell Systems Technical Journal, 7: 535–563. Kahraman, C., Cebeci, U., and Ruan, D., 2004, Multi-attribute comparison of catering service companies using fuzzy AHP: the case of Turkey, International Journal of Production Economics, 87: 171–184. K ulak, O., and Kahraman, C., 2005, Multi-attribute comparison of advanced manufac turing systems using fuzzy vs. crisp axiomatic design, International Journal of Production Economics, 95: 415–424 Kulak, O., Durmusoglu, M.B., and Kahraman, C., 2 005, Multi-attribute equipment selection based on information axiom, Journal of Materials Processing Technology, 169: 337–345. Nelson, C.A., 1986, A scoring model for flexible manufacturing systems project selection, European Journal of Opera tional Research, 24: 346–359. Shannon, C.E., 1948, The mathematical theory of comm unication, The Bell System Technical Journal, 27: 379–423. Suh, N.P., 2001, Axioma tic Design: Advances and Applications, Oxford University Press, New York. Suh, N .P., 1995, Design and operation of large systems, Annals of CIRP, 14(3): 203–213. Suh, N.P., 1990, The Principles of Design, Oxford University Press, New York. Za deh, L.A., 1965, Fuzzy sets, Information and Control, 8: 338–353.

MEASUREMENT OF LEVEL-OFSATISFACTION OF DECISION MAKER IN INTELLIGENT FUZZY-MCDM THEORY: A GENERALIZED APPROACH Pandian Vasant1, Arijit Bhattacharya2, and Ajith Abraham3 Electrical and Electronic Engineering Program, Universiti Teknologi Petronas, Tr onoh, BSI, Perak DR, Malaysia 2Embark Initiative Post-Doctoral Research Fellow, School of Mechanical & Manufacturing Engineering, Dublin City University, Glasne vin, Dublin 9, Ireland 3Center of Excellence for Quantifiable Quality of Service , Norwegian University of Science and Technology, Trondheim, Norway 1 Abstract: The earliest definitions of decision support systems (DSS) identify DSS as syste ms to support managerial decision makers in unstructured or semiunstructured dec ision situations. They are also defined as a computer-based information systems used to support decision-making activities in situations where it is not possibl e or not desirable to have an automated system perform the entire decision proce ss. This chapter aims to delineate measurement of level-of-satisfaction during d ecision making under an intelligent fuzzy environment. Before proceeding with th e multi-criteria decision making model (MCDM), authors try to build a co-relatio n among DSS, decision theories, and fuzziness of information. The co-relation sh ows the necessity of incorporating decision makers’ level-of-satisfaction in MCDM models. Later, the authors introduce an MCDM model incorporating different cost factor components and the said level-of-satisfaction parameter. In a later chapt er, the authors elucidate an application as well as validation of the devised mo del. The strength of the proposed MCDM methodology lies in combining both cardin al and ordinal information to get eclectic results from a complex, multi-person and multi-period problem hierarchically. Decision support system, level-of-satis faction in MCDM Key words: C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 235

236 P. Vasant et al. 1. INTRODUCTION Nomenclature D Decision matrix A Pair-wise comparison matrix among criteria (m × n ) m Number of criteria n Number of alternatives of the pair-wise comparison matr ix Principal eigen value of “A” matrix max PV Priority vector I.I. Inconsistency ind ex of “A” matrix R.I. Random inconsistency index of “A” matrix I.R. Inconsistency ratio of “A” matrix Level of satisfaction of decision maker OFM Objective factor measure S FM Subjective factor measure OFC Objective factor cost SI Selection index Fuzzy parameter that measures the degree of vagueness; = 0 indicates crisp. 1.1 DSS and Their Components Decision support systems (DSS) can be defined as computer-based information syst ems that aid a decision maker in making decisions for semi-structured problems. Numerous definitions to DSS exist. The earliest definitions of DSS (Gorry and Mo rton, 1977) identify DSS as systems to support managerial decision makers in uns tructured or semi-unstructured decision situations. Ginzberg and Stohr (1981) pr opose DSS as “a computer-based information system used to support decision making activities in situations where it is not possible or not desirable to have an au tomated system performs the entire decision process.” However, the most apt workin g definition is provided by Turban (1990). According to Turban (1990) “a DSS is an interactive, flexible, and adaptable computer based information system that uti lizes decision rules, models, and model base coupled with a comprehensive databa se and the decision maker’s own insights, leading to specific, implementable decis ions in solving problems that would not be amenable to management science models per se. Thus, a DSS supports complex decision making and increases its effectiv eness.” Alter (2004) explores the assumption that stripping the word system from D SS, focusing on decision support, and using ideas related to the work

Intelligent Fuzzy-MCDM Theory 237 system method might generate some interesting directions for research and practi ce. Some of these directions fit under the DSS umbrella, and some seem to be exc luded because they are not directly related to a technical artifact called a DSS . Alter (2004) suggests that “decision support is the use of any plausible compute rized or non-computerized means for improving sense making and/or decision makin g in a particular repetitive or non-repetitive business situation in a particula r organization.” However, the main objectives of DSS can be stated as follows: 1. To provide assistance to decision makers in situations that are semistructured, 2. To identify plans and potential actions to resolve problems, 3. To rank the s olutions identified that can be implemented and provide a list of viable alterna tives. DSS attempts to bring together and focus several independent disciplines. These are as follows: 1. 2. 3. 4. 5. 6. Operations research (OR), Management sc ience (MS), Database technology, Artificial intelligence (AI), Systems engineeri ng, Decision analysis. Artificial intelligence is a field of study that attempts to build software syst ems exhibiting near-human “intellectual” capabilities. Modern works on AI are focuse d on fuzzy logic, artificial neural networks (ANNs), and genetic algorithms (GAs ). These works, when integrated with DSS, enhance the performance of making deci sions. AI systems are used in creating intelligent models, analyzing models inte lligently, interpreting results found from models intelligently, and choosing mo dels appropriately for specific applications. Decision analysis may be divided i nto two major areas. The first, descriptive analysis, is concerned with understa nding how people actually make decisions. The second, normative analysis, attemp ts to prescribe how people should make decisions. Both are issues of concern to DSS. The central aim of decision analysis is improving decision making processes . Decisions, in general, are classified into three major categories: Structured decisions, Unstructured decisions, Semi-structured decisions.

238 P. Vasant et al. Structured decisions are those decisions where all steps of decision making are well structured. Computer code generation is comparatively easy for these types of decisions. In unstructured decisions, none of the steps of decision making is structured. AI systems are being built up to solve the problems of unstructured decisions. Semi-structured decisions comprise characteristics of structured and unstructured decisions. The DSS framework contains two types of components, whi ch may be used either individually or in tandem. The first component is a multio bjective programming (MOP) model, which employs mathematical programming to gene rate alternative mitigation plans. Typically, an MOP model must be formulated fo r the specific problem at hand, but once formulated, it can be solved on a compu ter using commercially available software. The second component is a multi-crite ria decision making (MCDM) model, used for evaluating decision alternatives that have been generated either by the MOP model or by some other method. MCDM model s are typically “shells” that can be applied to a wide range of problem types. A var iety of MCDM methodologies exist, some of which are available in the form of com mercial software. A manufacturing information system can also be used in conjunc tion with the DSS, for both managing data and for compiling decisions of alterna tive plans generated by the DSS. 1.2 Decision-Making Processes Strategic, tactical, and operative decisions are made on the various aspects of business operations. The vision of an industrial enterprise must take into consi deration the possible changes in its operational environment, strategies, and th e leadership practices. Decision making is supported by analyses, models, and co mputer-aided tools. Technological advances have an impact on the business of ind ustrial enterprises and on their uses of new innovations. Industrial innovations contribute to increased productivity and the diversification of production and products; they help to create better, more challenging jobs and to minimize risk s. Long-term decisions have an impact on process changes, functional procedures and maintenance and also on safety, performance, costs, human factors and organi sations. Short-term decisions deal with daily actions and their risks. Decisionmaking is facilitated by an analysis that incorporates a classification of one’s o wn views, calculating numerical values, translating

Intelligent Fuzzy-MCDM Theory 239 the results of analysis into concrete properties and a numerical evaluation of t he properties. One method applied for this purpose is the Analytic Hierarchy Pro cess (AHP) model (Saaty, 1990). This model, which has many features in common wi th the other MCDMs applied in the current research work, is suited for manufactu ring decision making processes that aim at making the correct choices in both th e short and the long term. 1.3 MCDM According to Agrell (1995) MCDM offers the methodology for decision making analy sis when dealing with multiple objectives. This may be the case when the success of the application depends on the properties of the system, the decision maker, and the problem. Problems with engineering design involve multiple criteria: th e transformation of resources into artifacts, a desire to maximize performance, and the need to comply with specifications. The MCDM methodology can be used to increase performance and to decrease manufacturing costs and delays of enterpris es. The MultipleCriteria Decision Support System (MC-DSS) uses the MCDM methodol ogy and ensures mathematical efficiency. The system employs graphical presentati ons and can be integrated with other design tools. Modeling and analyzing comple x systems always involve an array of computational and conceptual difficulties, whereas a traditional modeling approach is based primarily on simulation and con cepts taken from control theory. The strength of the MCDM lies in the systematic and quantitative framework it offers to support decision making. Comprehensive tuning or parametric design of a complex system requires elaboration on using th e modeling facilities of system dynamics and on the interactive decision making support of the MCDM. Most experienced decision makers do not rely on a theory to make their decisions because of cumbersome techniques involved in the process o f making decisions. But analytic decision making is of tremendous value when the said analytic process involves simple procedures and is accessible to the lay u ser as well as it possesses meaningful scientific justification of the highest o rder (Saaty, 1994).

240 P. Vasant et al. The benefits of descriptive analytical approaches for decision making are as fol lows (Saaty, 1994): 1. To permit decision makers to use information relating to decision making in a morphological way of thoroughly modeling the decision and t o make explicit decision makers’ tactical knowledge; 2. To permit decision makers to use judgments and observations in order to surmise relations and strengths of relations in the flow of interacting forces moving from the general to the part icular and to make predictions of most likely outcomes; 3. To enable decision ma kers to incorporate and trade off attribute values; 4. To enable decision makers to include judgments that result from intuition, day-to-day experiences, as wel l as those that result from logic; 5. To allow decision makers to make gradual a nd more thorough revisions and to combine the conclusions of different people st udying the same problem in different places. 1.4 Information vis-à-vis MCDM Theories Information is a system of knowledge that has been transformed from raw “data” into some meaningful form. Data are the raw materials for information. Data are also expressions of “events.” Information has value in current or prospective decision ma king at a specified time and place for taking appropriate “action” resulting in eval uation of “performance.” In this context attention is drawn to Figure 1. The terms “da ta” and “information” are often used interchangeably, but there is a distinction in th at. Data are processed to provide information, and the information is related to decision making (Davis, 1974). A schematic diagram illustrating relationship be tween data and information is shown in Figure 2. If there is no need for making decisions, information would be unnecessary. Information is the currency of the new economy. Yet most real-world cases lack the means to effectively organize an d distribute the information their employees need to make quick, smart business decisions. A structured, personalized, self-serve way to access information and collaborate across departmental and geographical boundaries provides the basic n eeds for making a good decision.

Intelligent Fuzzy-MCDM Theory 241 Figure 1. Generation and utilization of information Data Data Processing Information Figure 2. Converting raw data by an information system into useful information 1.5 1.5.1 Hidden Parameters in Information Uncertainty in Information Uncertainty permeates understanding of the real world. The purpose of informatio n systems is to model the real world. Hence, information systems must be able to deal with uncertainty. Many information systems include capabilities for dealin g with some kinds of uncertainty. For example, database systems can represent mi ssing values, information retrieval systems can match information to requests us ing a “weak” matching algorithm, and expert systems can represent rules that are kno wn to be true only for “most” or “some” of the time. By and large, commercial informatio n systems (e.g., database systems, information retrieval systems, or expert syst ems) have been slow to incorporate capabilities for dealing with uncertainty. Un certainty also has a long history of being associated with decision making resea rch as Harris (1998) notes: Decision making is the process of sufficiently reduc ing uncertainty and doubt about alternatives to allow a reasonable choice to be

242 P. Vasant et al. made from among them. This definition stresses the information gathering functio n of decision making. It should be noted here that uncertainty is reduced rather than eliminated. Very few decisions are made with absolute certainty because co mplete knowledge about all the alternatives is seldom possible. Researchers in v arious fields have also been concerned with the relationship between uncertainty and information seeking. In information science, the idea of uncertainty underl ies all aspects of information seeking and searching. Kuhlthau (1993) has propos ed uncertainty as a basic principle for information seeking, defining uncertaint y as “a cognitive state which commonly causes affective symptoms of anxiety and la ck of confidence.” And, drawing on her research, she notes that, “Uncertainty and an xiety can be expected in the early stages of the information search process.… Unce rtainty due to a lack of understanding, a gap in meaning, or a limited construct initiates the process of information seeking.” One of the biggest challenges for a manufacturing decision maker is the degree of uncertainty in the information t hat he or she has to process. In making some decisions, this is especially obvio us when experts in the same area provide conflicting opinions on the attributes meant for making decisions. Disagreement among experts making decisions results in conflicting effects information. The decision maker is likely to place increa sed importance on the source of the information. This in itself is not surprisin g, but the battle of the credentials that follows perhaps is. There seems to be a danger that the may come to rely on the reputation of an expert, rather than o n ensuring thorough scrutiny of the information that he or she has provided. Act ors in the decision making process may use uncertainty in the effects, and infor mation as a means to promote their attributes. A proponent can try to downplay t he effects of a development because they may not occur, whereas those in opposit ion may attempt to stall a project claiming that the disputed effects are likely to happen and are serious in nature. The decision maker is then left with the d ifficult task of navigating these disparities to come to a decision. In particul ar in the face of uncertainty, there seems to be a human tendency to make person al observations the deciding factor. 1.5.1.1 Sources of Uncertainty Uncertaintie s are solely due to the unavailability of “perfect” information. Uncertainty might r esult from using unreliable information sources, for example, faulty reading ins truments, or input forms that have been filled

Intelligent Fuzzy-MCDM Theory 243 out incorrectly (intentionally or inadvertently). In other cases, uncertainty is a result of system errors, including input errors, transmission “noise,” delays in processing update transactions, imperfections of the system software, and corrup ted data owing to failure or sabotage. At times, uncertainty is the unavoidable result of information gathering methods that require estimation or judgment. In other cases, uncertainty is the result of restrictions imposed by the model. For example, if the database schema permits storing at most two occupations per emp loyee, descriptions of occupation would exhibit uncertainty. Similarly, the shee r volume of information that is necessary to describe a real-world object might force the modeler to turn to approximation and sampling techniques. 1.5.1.2 Degr ee of Uncertainty The relevant information that is available in the absence of c ertain information may take different forms, each exhibiting a different level o f uncertainty. Uncertainty is highest when the mere existence of some real-world object is in doubt. The simplest solution is to ignore such objects altogether. This solution, however, is unacceptable if the model claims to be closed world (i.e., objects not modeled do not exist). Uncertainty is reduced somewhat when e ach element is assigned a value in a prescribed range, to indicate the certainty that the modeled object exists. When the element is a fact, this value can be i nterpreted as the confidence that the fact holds; when it is a rule, this value can be interpreted as the strength of the rule (percent of cases where the rule applies). Now it is assumed that “existence” is assured, but some or all of the info rmation with which the model describes an object is unknown. Such information ha s also been referred to as incomplete, missing, or unavailable. Uncertainty is r educed when the information that describes an object is known to come from a lim ited set of alternatives (possibly a range of values). This uncertainty is refer red to as disjunctive information. Note that when the set of alternatives is sim ply the entire “universe,” this case reverts to the previous (less informative) case . Uncertainty is reduced even more when each alternative is accompanied by a num ber describing the probability that it is indeed the true description (and the s um of these numbers for the entire set is 1). In this case, the uncertain inform ation is probabilistic. Again, when the probabilities are unavailable, probabili stic information becomes disjunctive information.

244 P. Vasant et al. Occasionally, the information available to describe an object is descriptive rat her than quantitative. Such information is often referred to as fuzzy or vague i nformation. 1.5.1.3 Vagueness in Information Russell (1923) attributes vagueness to being mostly a problem of language. Of course, language is part of the probl em, but it is not the main problem. There would still be vagueness even if we ha d a very precise, logically structured, language. The principal source of vaguen ess seems to be in making discreet statements about continuous phenomenon. Accor ding to Russell (1923), “Vagueness in a cognitive occurrence is a characteristic o f its relation to that which is known, not a characteristic of the occurrence in itself.” Russell (1923) adds, “Vagueness, though it applies primarily to what is co gnitive, is a conception applicable to every kind of representation.” Surprisingly , Wells (1908) was among the first to suggest the concept of vagueness: Every sp ecies is vague, every term goes cloudy at its edges, and so in my way of thinkin g, relentless logic is only another name for stupidity for a sort of intellectua l pigheadedness. If you push a philosophical or metaphysical enquiry through a s eries of valid syllogisms never committing any generally recognized fallacy you nevertheless leave behind you at each step a certain rubbing and marginal loss o f objective truth and you get deflections that are difficult to trace, at each p hase in the process. Every species waggles about in its definition, every tool i s a little loose in its handle, every scale has its individual. In real-world pr oblems there is always a chance of getting introduced to the vagueness factor wh en information deals in combination with both cardinal and ordinal measures. It should always be remembered that reduction of vagueness is to be addressed in a situation where decision alternatives are well inter-related and have both cardi nal and ordinal criteria for selection. 1.5.1.4 Sources of Vagueness Linguistic expressions in classic decision making processes incorporate unquantifiable, imp erfect, nonobtainable information and partially ignorant facts. Data combining b oth ordinal and cardinal preferences in real-world decision making problems are highly unreliable and both contain a certain degree of vagueness. Crisp data oft en contains some amount of vagueness

Intelligent Fuzzy-MCDM Theory 245 and, therefore, need the attention of decision makers in order to achieve a less er degree of vagueness inherent. The purpose of decision making processes is bes t served when imprecision is communicated as precisely as possible but no more p recisely than warranted. 2. PRIOR WORKS ON FUZZY-MCDM FOR SELECTING BEST CANDIDATEALTERNATIVE The available literature on MCDM tackling fuzziness is as broad as it is diverse . Literature contains several proposals on how to incorporate the inherent uncer tainty as well as the vagueness associated with the decision maker’s knowledge int o the model (Arbel, 1989; Arbel and Vargas, 1990; Banuelas and Antony, 2004; Saa ty and Vargas, 1987). The analytic hierarchy process (AHP) (Saaty, 1980 and 1990 ) literature, in this regard, is also vast. There has been a great deal of inter est in the application of fuzzy sets to the representation of fuzziness and unce rtainty in management decision models (Buckley, 1988; Chen and Hwang, 1982; Ghot b and Warren, 1995; Gogus and Boucher, 1997; Van Laarhoven and Pedrycz, 1983; Li ang and Wang, 1994; Lai and Hwang, 1994; Zimmerman, 1976, 1987). Some approaches were made to handle the uncertainties of MCDM problems. Bellman and Zadeh (1970 ) have shown fuzzy set theory’s applicability to the MCDM study. Yager and Basson (1975) and Bass and Kwakernaak (1977) have introduced maximin and simple additiv e weighing model using the membership function (MF) of the fuzzy set. Most of th e recent literature is filled with mathematical proofs. A decision maker needs a n MCDM assessment technique in regard to its fuzziness that can be easily used i n practice. An approach was taken earlier by Marcelloni and Aksit (2001). Their aim was to model inconsistencies through the application of fuzzy logic-based te chniques. Boucher and Gogus (2002) examined certain characteristics of judgment elicitation instruments appropriate to fuzzy MCDM. In their work the fuzziness w as measured using a gamma function. By defining a decision maker’s preference stru cture in fuzzy linear constraint (FLC) with soft inequality, one can operate the concerned fuzzy optimization model with a modified S-curve smooth MF to achieve the desired solution (Watada, 1997). One form of logistic MF to overcome

246 P. Vasant et al. difficulties in using a linear membership function in solving a fuzzy decision m aking problem was proposed by Watada (1997). However, it is expected that a new form of logistic membership function based on nonlinear properties can be derive d, and its flexibility in fitting real-life problem parameters can be investigat ed. Such a formulation of a nonlinear logistic MF was presented in this work, an d its flexibility in taking up the fuzziness of the parameter in a real-life pro blem was demonstrated. Carlsson and Korhonen (1986) have illustrated, through an example, the usefulness of a formulated MF, viz., an exponential logistic funct ion. Their illustrated example was adopted to test and compare a nonlinear MF (L ootsma, 1997). Such an attempt using the said validated nonlinear MF and compari ng the results was made by Vasant et al. (2005). Comprehensive tests based on a real-life industrial problem have to be undertaken on the newly developed member ship function in order to prove further its applicability in fuzzy decision maki ng (Vasant, 2003; Vasant et al., 2002; 2005). To test the newly formulated MF in problems as stated above, a software platform is essential. In this work MATLAB has been chosen as the software platform using its M-file for greater flexibili ty. In the past, studies on decision making problems were considered on the bipa rtite relationship of the decision maker and analyst (Tabucanon, 1996). This is with the assumption that the implementers are a group of robots that are program med to follow instructions from the decision maker. This notion is now outdated. Now a tripartite relationship is to be considered, as shown on Figure 3, where the decision maker, the analyst, and the implementer will interact in finding a fuzzy satisfactory solution in any given fuzzy system. This is because the imple menters are human beings, and they have to accept the solutions given by the dec ision maker to be implemented under a turbulent environment. In case of triparti te fuzzy systems, the decision maker will communicate and describe the fuzzy pro blem with an analyst. Based on Figure 3. Tripartite relationship for MCDM problems

Intelligent Fuzzy-MCDM Theory 247 the data that are provided by the decision maker, the analyst will formulate MFs , solve the fuzzy problems, and provide the solution back to the decision-maker. After that, the decision maker will provide the fuzzy solution with a trade off to the implementer for implementation. An implementer has to interact with deci sion maker to obtain an efficient and highly productive fuzzy solution with a ce rtain degree of satisfaction. This fuzzy system will eventually be called a high productive fuzzy system (Rommelfanger, 1996). A tripartite relationship, decisi on maker analyst implementer, is essential to solve any industrial problem. The following criticisms of the existing literatures, in general, are made after a s tudy of the existing vast literature on the use of various types of MFs in findi ng out fuzziness patterns of MCDM methodologies: 1. Data combining both ordinal and cardinal preferences contain nonobtainable information and partially ignoran t facts. Both ordinal and cardinal preferences contain a certain degree of fuzzi ness and are highly unreliable, unquantifiable and imperfect. 2. Simplified fuzz y MFs, viz., trapezoidal and triangular and even gamma functions, are not able t o bring out real-world fuzziness patterns in order to elucidate a degree of fuzz iness inherent in the MCDM model. 3. Level-of-satisfaction of the decision maker s should be judged through a simple procedure while making decisions through MCD M models. 4. An intelligent tripartite relationship among the decision maker, an alyst and implementer is essential, in conjunction to a more flexible MF design, to solve any real-world MCDM problem. Among many diversified objectives of the current work, one objective is to find out fuzziness patterns of the candidate-a lternatives having disparate level-of-satisfaction in MCDM model. Relationships among the degree of fuzziness, level-of-satisfaction and the selection-indices o f the MCDM model guide decision makers under a tripartite fuzzy environment in o btaining their choice tradeoff with a predetermined allowable imprecision. Anoth er objective of the current work is to provide a robust, quantified monitor of t he level-of-satisfaction among decision makers and to calibrate these levels of satisfaction against decision makers’ expectations. Yet another objective is to pr ovide a practical tool for further assessing the impact of different options and available courses of action.

248 P. Vasant et al. 3. COMPONENTS OF THE MCDM MODEL The proposed MCDM model considers a fuzziness pattern in disparate level-of-sati sfaction of the decision maker. The model outlines a MF for evaluating degree of fuzziness hidden in the Eq. (1). AHP provides the decision maker’s with a vector of priorities (PV) to estimate the expected utilities of each candidate-FMS. A m athematical model was proposed by Bhattacharya et al. (2004, 2005) to combine co st factor components with the importance weightings found from AHP. The governin g Eq. of the said model is: OFM = Objective factor measure, OFC = Objective fact or cost, SFM = Subjective factor measure, SI = Selection index, = Objective fact or decision weight, n = Finite number of candidate-alternative. SI i SFM i 1 OFM i (1) where OFM l OFC l 1 n OFC l (2) l 1 In the said model, AHP plays a crucial role. AHP is an MCDM method, and it refer s to making decisions in the presence of multiple, usually conflicting, criteria . A criterion is a measure of effectiveness. It is the basis for evaluation. Cri teria emerge as a form of attributes or objectives in the actual problem setting . In reality, multiple criteria usually conflict with each other. Each objective /attribute has a different unit of measurement. Solutions to the problems by AHP are either to design the best alternative or to select the best one among the p reviously specified finite alternatives. For assigning the weights to each of th e attributes as well as to the alternative processes for constructing the decisi on matrix and pair-wise comparison matrices, the phrase like “much more important” i s used to extract the decision maker’s preferences. Saaty (1990) gives an intensit y scale of importance (refer to Table 1) and has broken down the importance rank s.

Intelligent Fuzzy-MCDM Theory Table 1. The Nine-Point Scale of Pair-Wise Comparison Intensity scale 1 3 5 7 9 2, 4, 6, 8 Interpretation Equally important Moderately preferred Essentially pre ferred Very strongly preferred Extremely preferred Intermediate importance betwe en two adjacent judgments 249 In AHP the decision matrix is always a square matrix. Using the advantage of pro perties of eigenvalues and eigenvectors of a square matrix, the level of inconsi stency of the judgmental values assigned to each elements of the matrix is check ed. In this chapter the proposed methodology is applied to calculate the priorit y weights for functional, design factors and other important attributes by eigen vector method for each pair-wise comparison matrix. Next, global priorities of v arious attributes rating are found by using AHP. These global priority values ar e used as SFM in Eq. (1). The pair-wise comparison matrices for five different f actors are constructed on the basis of Saaty’s nine-point scale (refer to Table 1) . The objective factors, i.e., OFM, and OFC are calculated separately by using c ost factor components. In the mathematical modeling for finding the SFMi values, decomposition of the total problem (factor-wise) into smaller sub-problems has been done. This is done so that each sub-problem can be analyzed and appropriate ly handled with practical perspectives in terms of data and information. The obj ective of decomposition of the total problem for finding out the SFM values is t o enable a pair-wise comparison of all the elements on a given level with respec t to the related elements in the level just above. The proposed algorithm consis ts of a few steps of calculations. Prior to the calculation part, listing of the set of candidate-alternatives is carried out. Next, the cost components of the candidate-alternatives are quantified. Factors, on which the decision making is based, are identified as intrinsic and extrinsic. A graphical representation dep icting the hierarchy of the problem in terms of overall objective, factors, and number of alternatives is to be developed. Next follows the assigning of the jud gmental values to the factors as well as to the candidate-alternatives to constr uct the decision matrix and pair-wise comparison matrices, respectively. A decis ion matrix is constructed by assigning weights to each factor based on the relat ive importance of its contribution according to a ninepoint scale (refer to Tabl e 1). Assigning the weights to each candidate-

250 P. Vasant et al. alternative for each factor follows the same logic as that of the decision matri x. This matrix is known as a pair-wise comparison matrix. The PV values are dete rmined then for both the decision and the pair-wise comparison matrices. The max for each matrix may be found by multiplication of the sum of each column with t he corresponding PV value and subsequent summation of these products. There is a “check” in the judgmental values given to the decision and pair-wise comparison mat rices for revising and improving the judgments. If I.R. is greater than 10%, the values assigned to each element of the decision and pair-wise comparison matric es are said to be inconsistent. For I.R. < 10%, the level of inconsistency is ac ceptable. Otherwise the level of inconsistency in the matrices is high and the d ecision maker is advised to revise the judgmental values of the matrices to prod uce more consistent matrices. It is expected that all the comparison matrices sh ould be consistent. But the very root of the judgment in constructing these matr ices is the human being. So, some degree of inconsistency of the judgments of th ese matrices is fixed at 10%. Calculation of I.R. involves I.I., R.I., and I.R. These matrices are evaluated from Eqs. (3), (4) and (5) respectively. I.I. = ( max (n n) 1) (3) R.I. = [1.98 (n n I.I. R.I. 2)] (4) I.R. = (5) The OFMi values are determined by Eq (6). n OFM i [OFCi i 1 1 ] OFCi 1 (6) The SFMi values are the global priorities for each candidate-alternative. SFMi m ay be found by multiplying each of the decision matrix PV values to each of the

PV value of each candidate-alternative for each factor. Each product is then sum med up for each alternative to get SFMi. For an easy demonstration of the propos ed fuzzified MCDM model, efforts for additional fuzzification are confined assum ing that differences in judgmental values are only 5%. Therefore, the upper boun d and lower

Intelligent Fuzzy-MCDM Theory 251 bound of SFMi as well as SIi indices are to be computed within a range of 5% of the original values. In order to avoid complexity in delineating the technique p roposed hereinbefore, we have considered the 5% measurement. One can fuzzify the SFMi values from the very beginning of the model by introducing a modified S-cu rve MF in AHP, and the corresponding fuzzification of SIi indices can also be ca rried out using the holistic approach used in Eq. (1). The set of candidate-alte rnatives are then ranked according to the descending order of SIi indices (refer to Eq. 7). ~ LSI i SFM i LSI L LSIU LSI L ln 1 C A LSI i 1 (7) In this work, a monotonically nonincreasing logistic function has been used as a membership function: f ( x) B 1 Ce x (8) where is the level-of-satisfaction of the decision maker; B and C are is a fuzzy parameter that measures the scalar constants; and , 0< < degree of vagueness (f uzziness), wherein = 0 indicates crisp. Fuzziness . becomes highest when The gen eralized logistic membership function is defined as 1 f ( x) B 1 Ce 0 x x xL x xL x xU xU (9) To fit into the MCDM model in order to sense its degree of fuzziness, the Eq. (9 ) is modified and redefined as follows:

1 0.999 B 1 Ce 0.001 0 x x x x x x a x a x x x xb b a x x b (10) (x) In Eq. (10) the membership function is redefined as 0.001 0.999. This range is selected because in real-world situations the workforce need not be always 10 0% of the requirement. At the same time

252 P. Vasant et al. the workforce will not be 0%. Therefore, there is a range between x0 and x1 ( x) 0.999. This concept of range of ( x) is used in this with 0.001 chapter. Choice of the level-of-satisfaction of the decision maker, i.e., , is an important iss ue. It is the outcome of the aggregate decision by the design engineer, producti on engineer, maintenance engineer, and capital investor of a manufacturing organ ization. However, the selection of a candidatealternative may give different set s of results for different values of for the same attributes and cost factor com ponents. That’s why the proposed model includes fuzzy-sensitivity plots to analyse the effect of as well as the degree of fuzziness, , in the candidate-alternativ e selection problem. 4. 4.1 FORMULATION OF THE INTELLIGENT FUZZIFIED MCDM MODEL Membership Function There are 11 in-built membership functions in the MATLAB fuzzy toolbox. In the c urrent study, a modified version of No. 7 MF has been used. All the built-in MF includes 0 and 1. In the current work, 0 and 1 have been excluded and the S-shap ed membership function has been extensively modified accordingly. As mentioned b y Watada (1997), a trapezoidal MF will have some difficulties such as degenerati on, i.e., some sort of deterioration of solution, while introducing fuzzy proble ms. In order to solve the issue of degeneration, we should employ a non linear l ogistic function such as a tangent hyperbolic that has asymptotes at 1 and 0. In the current work, we employ the logistic function for the nonlinear membership function as given by f ( x) B 1 Ce x (11) < is a fuzzy where B and C are scalar constants and , 0 < parameter that measure s the degree of vagueness, wherein = 0 indicates . crisp. Fuzziness becomes high est when Eq. (11) will be of the form as indicated by Figure 4 when 0 < < .

Intelligent Fuzzy-MCDM Theory 253 Figure 4. Variation of logistic MF with respect to fuzzy parameter, (where m2 > m1) The reason why we use this function is that the logistic MF has a similar shape as that of the tangent hyperbolic function employed by Leberling (1981) but it i s more flexible (Bells, 1999) than the tangent hyperbola. It is also known that a trapezoidal MF is an approximation to a logistic function. Therefore, the logi stic function is very much considered an appropriate function to represent a vag ue goal level. This function is found to be very useful in making decisions and in implementation by the decision maker and implementer (Lootsma, 1997; Zimmerma n, 1985; 1987). Moreover, to avoid linearity in the real-life application proble ms, especially in industrial engineering problems, a non linear function such as modified MF can be used. This MF is used when the problems and its solutions ar e independent (Varela and Riberio, 2003). It should be emphasized that some nonl inear MFs such as S-curve MFs are much more desirable for real-life application problems than that of linear MFs. The logistic function, Eq. (11), is a monotoni cally nonincreasing function, which will be employed as a fuzzy MF. This is very important because, due to an uncertain environment the availability of the vari ables are represented by the degree of fuzziness. The said MF can be shown to be non increasing as df dx BC e x (1 Ce x )2 (12)

254 P. Vasant et al. An MF is flexible when it has vertical tangency, an inflexion point, and asympto tes. Since B, C, , and x are all greater than zero, df dx 0. Furthermore it can be shown that Eq. (11) has asymptotes at f(x) = 0 and f(x) = 1 at appropriate values of B and C. This implies: – lim x df dx 0 and lim x df 0 dx 0 These asymptotes can be proved as follows. From Eq. (12), one gets lim x df dx . Therefore, using L’hopital’s rule, one obtains lim x df dx lim x B =0 2(1 Ce x ) (13) 0 , the situation is less vague and hence As x From Eq. (12), one gets lim x 0 0. df dx BC (1 C ) 2 0 , when 0

(14) In addition to the above equation, it can be shown that the logistic function Eq . (11) has a vertical tangent at x = x0 , x0 is the point where f(x0) = 0.5. Fur thermore it can also be shown that the said logistic function has a , with f ( x) being the point of inflexion at x = x0, such that f ( x0 ) second deriva tive of f(x) with respect to x. An MF of S-curve nature, in contrast to linear f unction, exhibits the real-life problem. The generalized logistic MF is defined as

Intelligent Fuzzy-MCDM Theory 255 1 f ( x) B 1 Ce 0 x x xL x xL x xU xU (15) The S-curve MF is a particular case of the logistic function defined in Eq. (15) . The said S-curve MF has specific values of B, C and . The logistic function as defined in Eq. (11) was indicated as an S-curve MF by Zadeh (1971; 1975). 4.2 Design of Modified, Flexible S-curve MF To fit into the MCDM model in order to sense its degree of fuzziness, Eq. (15) i s modified and redefined as follows and illustrated in Figure 5. 1 0.999 x B 1 Ce 0.001 0 ( x) x x x xa x x xa xa x xb xb xb (16) 1.0 0.999 0.5 0.001 0 x a x0 x b x Figure 5. Modified S-curve membership function

256 P. Vasant et al. We rescale the x-axis as xa = 0 and xb = 1 in order to find the values of B, C, and . Nowakowska (1977) has performed such a rescaling in his work on the social sciences. The values of B, C, and are obtained from Eq. (16) as B = 0.999 (1 + C) (17) B 1 Ce y 0.001. (18) By substituting Eq. (17) into Eq. (18), one gets 0.999 1 C 1 Ce y Rearranging Eq. (19), one gets 0.001. (19) ln 1 0.998 0.999 . 0.001 C (20) Since B and depend on C, one requires one more condition to get the values for B , C, and . Let, when x0 Therefore, xa 2 xb , (x0) = 0.5. B 1 Ce and hence 2 0.5 , (21) 2 ln 2B 1 . C (22)

Intelligent Fuzzy-MCDM Theory 257 Substituting Eq. (20) and Eq. (21) into Eq. (22), we obtain 2 ln 2(0.999)(1 C ) 1 C 2 ln 1 0.001 0.998 C 0.999 (23) 0.998 1.998C C 998 999C which in turn yields (24) Eq. (24) is solved and it is found that C –994.011992 988059.8402 1990.015992 3964.127776 (25) Since C has to be positive, Eq. (22) gives C 0.001001001 , and from Eqs. (17) an d (22), one gets B = 1 and = 13.81350956. Thus, it is evident from the preceding sections that the flexible, modified S-curve MF can be more easily handled than other nonlinear MFs such as the tangent hyperbola. The linear MF such as the tr apezoidal MF is an approximation from a logistic MF and is based on many idealis tic assumptions. These assumptions contradict the realistic real-world problems. Therefore, the S-curve MF is considered to have more suitability in sensing the degree of fuzziness in the fuzzy-uncertain judgmental values of a decision make r. The modified S-curve MF changes its shape according to the fuzzy judgmental v alues of a decision maker and therefore, a decision maker finds it suitable to a pply his/her strategy to MCDM problems using these judgmental values. Thus the p roposed S-shaped membership function is flexible due to its following characteri stics: (i) (ii) (iii) (x) is continuous and strictly monotonously nonincreasing; (x) has lower and upper asymptotes at (x) = 0 and (x) = 1 as x and x 0, respect ively; (x) has inflection point at x0 1 ln 2 1 with A 1 C C

258 P. Vasant et al. The fuzzy intelligence of the proposed MCDM model is incorporated under a tripar tite environment. A fuzzy rule-based decision (if then rule) is incorporated in the algorithm to sense the fuzziness patterns under a disparate level-of-satisfa ction of the decision maker. The aim is to produce a rule that works well on pre viously unseen data. In the next chapter it will be demonstrated how to compute the degree of fuzziness and level-of-satisfaction, and a correlation among degre e of fuzziness having a disparate level-of-satisfaction and the selection indice s will also be elucidated to guide the decision maker in selecting the best cand idate-alternative under an unstructured environment. 5. CONCLUSION The proposed MCDM model shows how to measure a parameter called “level-of-satisfac tion” of the decision maker while making any kind of decision. “Level-of-satisfactio n” is a much-quoted terminology in classic as well as modern economics. To date, w e are not aware of any reported research work in which level-of-satisfaction has been measured with a rigorous mathematical logic. The proposed model is a one-o f-a-kind solution to incorporate the “level-of-satisfaction” of decision maker. Anot her solution can also be made with many sophisticated tools, like some approxima tion tool using neuro-fuzzy hybrid models. The strength of the proposed MCDM met hodology lies in combining both cardinal and ordinal information to get eclectic results from a complex, multi-person, and multi-period problem hierarchically. The methodology proposed in this chapter is very useful in quantifying the intan gible factors in a good manner and in finding out the best among the alternative s depending on their cost factors. Contrary to the traditional way of selection using discounted cash flow (DCF), this methodology is a sound alternative to app ly under an unstructured environment. There may be some weaknesses due to the no navailability of experts’ comments, i.e., judgmental values. Comparison among vari ous similar types of systems is the opportunity of the proposed model. An underl ying threat is associated with the proposed model that a illogical decisions and mis-presentation of experts comments may lead to a wrong decision. The MCDM met hodology proposed in this chapter assumes that the decision is made under a fuzz y environment. A comparative study by accommodating different measures of uncert ainty and risk in the MADM methodology may also be made to judge the best-suited measure of

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FMS SELECTION UNDER DISPARATE LEVELOF-SATISFACTION OF DECISION MAKING USING AN I NTELLIGENT FUZZY-MCDM MODEL Arijit Bhattacharya1, Ajith Abraham2, and Pandian Vasant3 1 Embark Initiative Post-Doctoral Research Fellow, School of Mechanical & Manufact uring Engineering, Dublin City University, Glasnevin, Dublin 9, Ireland 2Center of Excellence for Quantifiable Quality of Service, Norwegian University of Scien ce and Technology, Trondheim, Norway 3Electrical and Electronic Engineering Prog ram, Universiti Teknologi Petronas, Tronoh, BSI, Perak DR, Malaysia Abstract: This chapter outlines an intelligent fuzzy multi-criteria decision-making (MCDM) model for appropriate selection of a flexible manufacturing system (FMS) in a c onflicting criteria environment. A holistic methodology has been developed for f inding out the “optimal FMS” from a set of candidate-FMSs. This method of trade-offs among various parameters, viz., design parameters, economic considerations, etc ., affecting the FMS selection process in an MCDM environment. The proposed meth od calculates the global priority values (GP) for functional, design factors and other important attributes by an eigenvector method of a pair-wise comparison. These GPs are used as subjective factor measures (SFMs) in determining the selec tion index (SI). The proposed fuzzified methodology is equipped with the capabil ity of determining changes in the FMS selection process that results from making changes in the parameters of the model. The model achieves balancing among crit eria. Relationships among the degree of fuzziness, level-of-satisfaction and the SIs of the MCDM methodology guide decision makers under a tripartite fuzzy envi ronment in selecting their choice of trading-off with a predetermined allowable fuzziness. The measurement of level-of-satisfaction during making the appropriat e selection of FMS is carried out. FMS, intelligent fuzzy MCDM, global priority, sensitivity analysis, selection indices Key words: C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 263

264 A. Bhattacharya et al. 1. INTRODUCTION A flexible manufacturing system (FMS) is a set of integrated computercontrolled, automated material handling equipments and numerical-controlled machine tools c apable of processing a variety of part types. Due to the competitive advantages like flexibility, speed of response, quality, reduction of lead-time, reduction of labour etc., FMSs are now gaining popularity in industries. Today’s manufacturi ng strategy is purely a choice of alternatives. The better the choice, more will be the productivity as well as the profit maintaining quality of product and re sponsiveness to customers. In this era of rapid globalization, the overall objec tive is to purchase a minimum amount of capacity (i.e., capital investment) and utilize it in the most effective way. Although FMS is an outgrowth of existing m anufacturing technologies, its selection is not often studied. It has been a foc al point in manufacturing related research since the early 1970s. FMS provides a low inventory environment with unbalanced operations unique to the conventional production environment. The process design of an FMS consists of a set of cruci al decisions that are to be made carefully. It requires decision making, e.g., s election of a CNC machine tool, material handling system, or product mix. The se lection of a FMS thus requires trading-off among the various parameters of the F MS alternatives. The selection parameters are conflicting in nature. High-qualit y management is not enough for dealing with the complex and ill-structured facto rs that are conflicting in nature (Buffa, 1993). Therefore, there is a need for sophisticated and applicable technique to help the decision makers for selecting the proper FMS in a manufacturing organization. The authors, thus, propose a DS S methodology, for appropriate FMS selection, that trade off among some intangib le criteria as well as cost factors to get the maximum benefit out of these conf licting-in-nature criteria. There have been many contributors to the literature on selection of proper FMS. A selective review of some of the relevant works in this area is give here. Kaighobadi and Venkatesh (1994) presented an overview an d survey of research in FMSs. They also presented a definition of FMSs. Chen et al. (1998) investigated the relationship between flexibility measurements and sy stem performance in the flexible manufacturing systems environment. The authors suggested several alternative measures for the assessment of machine flexibility and routing flexibility—two of the most important flexibility types discussed in the literature. Nagarur (1992) showed that computer integration and flexibility of the system were the two critical factors of FMS. Eight different types of fle xibility were proposed by Browne et al. (1984). Each of these flexibilities cont ributes to overall flexibility of

FMS Selection Using a Fuzzy-MCDM Model 265 the system to cope with possible changes in demand structure. In addition to mac hine, process, product, routing, volume, expansion, operation and production fle xibility as described by Browne et al. (1984). Barad and Sipper (1988) introduce d another classification, i.e., transfer flexibility. Buzacott and Mandelbaum (1 985) defined flexibility as the ability of a manufacturing system to cope with c hanging circumstances. High-level flexibility enables a manufacturing firm to pr ovide faster response to market changes maintaining high product quality standar ds (Gupta and Goyal, 1989). Flexible manufacturing provides an environment where integration effects cannot be eliminated (Lenz, 1988). If the inventory is rais ed, the manufacturing environment becomes that of the job shop type. On the cont rary, if the operations are balanced, the environment becomes that of the transf er line. The changes in production are related to both inventory changes as well as changes in flow time. Three variables determine the amount of integration ef fects that result in a production process. These are inventory level, balanced l oading, and flexibility. Inventory level is quantified by counting the number of parts that are active in the production process. Balanced loading can be quanti fied by the use of flow time. The use of flow time is to measure the balance wit hin a production facility, and it is derived from the transfer line. The flow ti me provides a means to measure the balance between station loads in any type of production facility. Flexibility can be measured by the variability of the flow time. A process with greater degree of flexibility will provide less variability to the flow time. Meredith and Suresh (1986) addressed justification of economi c analysis and of analytical and strategic approaches in advanced manufacturing technologies. Evaluation of FMS alternatives was earlier carried out by Miltenbu rg and Krinsky (1987). They analyzed traditional economic evaluation techniques for the evaluation. Nelson (1986) formulated a scoring model for FMS project sel ection. Performance measures, viz., quality and flexibility, were also quantifie d in the scoring model. Use of the analytic hierarchy process (AHP) for evaluati on of tangible and intangible benefits during FMS investment was reported by Wab alickis (1988). Stam and Kuula (1991) developed a two-phase decision support pro cedure using AHP and multi-objective mathematical programming for selection of F MS. Sambasivarao and Deshmukh (1997) presented a DSS integrating multiattribute analysis, economic analysis and risk evaluation analysis. They have suggested AH P, TOPSIS (technique for order preference for similarity to ideal solution), and a linear additive utility model as an alternative multiattribute analysis model . Shang and Sueyoshi (1995) formulated a model of simulation and data envelopmen t analysis (DEA) along with AHP for FMS selection. Karsak and Tolga (2001) propo sed a fuzzy-MCDM approach for

266 A. Bhattacharya et al. evaluation of advanced manufacturing system investments considering economic and strategic selection criteria. Karsak (2002) proposed a robust decision-making p rocedure for evaluating FMS using a distance-based fuzzy-MCDM philosophy. Some r esearchers (Chen et al., 1998; Evans and Brown, 1989) believe that qualitative b enefits cannot be considered mathematically unless one uses a knowledge-based sy stem. This dissertation outlines a mathematical approach based on the judgmental values of a decision maker that can help decision makers in selecting the costeffective FMS. Abdel-Malek and Wolf (1991) propose a “measure” for the decisionmakin g process. The said “measure” ranks different competing FMS designs according to the ir inherent flexibility as they relate to the maximum flexibility possible stipu lated by the state-of-the-art. In developing the proposed “measure,” the attributes governing the flexibility of FMS major components are defined. A notion of “string s” representing alternative production routes for different products is set forth. The method allows the integration of the eight points of flexibility stated by Browne et al. (1984) into a single comprehensive flexibility indicator. Elango a nd Meinhart (1994) provide a framework for selection of an appropriate FMS using a holistic approach. The selection process considers operational and financial aspects. Furthermore, their selection process is consistent with industry, marke t, organizational, and other strategic needs. A DSS for dynamic task allocation in a distributed structure for flexible manufacturing systems FMS has been devel oped by Trentesaux et al. (1998). An entity of the manufacturing system is consi dered as an autonomous agent, called the integrated management station (IMS), ab le to cooperate with other agents to achieve a global production program. Cooper ation is performed by exchanging messages among the different agents. The charac teristics of a DSS that supports multi-criteria algorithms and sensitivity tests is presented in Trentesaux et al. (1998). This DSS is integrated to each decisi on system of every IMS. Trentesaux et al.’s (1998) research work aims at allocatin g tasks in a dynamic way by proposing to the human operator a selection of possi ble resources. Sarkis and Talluri (1999) disclose a model for evaluating alterna tive FMSs by considering both quantitative and qualitative factors. The evaluati on process uses a DEA model, which incorporates both ordinal and cardinal measur es. The model provides pair-wise comparisons of specific alternatives for FMSs. The consideration of both tangible and intangible factors is achieved in their m ethodology. The analysis of results provides both seller’s and buyer’s perspectives of FMS evaluation. The decision-making process for machine-tool selection and op eration allocation in a FMS usually involves multiple conflicting objectives. Ra i et al.

FMS Selection Using a Fuzzy-MCDM Model 267 (2002) address application of a fuzzy goal-programming concept to model the prob lem of machine-tool selection and operation allocation with explicit considerati ons given to objectives of minimizing the total cost of machining operation, mat erial handling, and set up. The constraints pertaining to the capacity of machin es, tool magazine, and tool life are included in the model. A genetic algorithm (GA)-based approach is adopted to optimize this fuzzy goal-programming model. Ad vanced computing/communications technology is present in virtually all areas of manufacturing. In the near future, a totally computer-controlled manufacturing e nvironmental will be a realistic expectation (Haddock and Hartshorn, 1989). The integration and enhancement of both computer-aided design (CAD) and computer-aid ed manufacturing (CAM) represents the foundations for achieving a totally integr ated manufacturing system. The requirements for increased responsiveness to mark et and the demands for shorter product introduction times underline the need for a coherent formal approach toward equipment selection to support the knowledge and experience of the engineers entrusted with this important task (Gindy and Ra tchev, 1998). With the increasing complexity of the decision making in manufactu ring system design, the search for the right structure depends on the capability of the designers to compare different solutions using common approaches in an i ntegrated decision-making environment (Gindy and Ratchev, 1998). Thus, machine t ool selection has strategic implications that contribute to the manufacturing st rategy of a manufacturing organization (Yurdakul, 2004). In such a case, it is i mportant to identify and model the links between machine tool alternatives and m anufacturing strategy (Yurdakul, 2004). Haddock and Hartshorn (1989) present a D SS that assists in the specific selection of a machine required to process speci fic dimensions of a part. The selection will depend on part characteristics, whi ch are labeled in a part code and correlated with machine specifications and qua lifications. The choice of the optimal machine, versus possible alternates, is m ade by a planner comparing a criterion measure. Some possible criteria for selec tion as suggested by Haddock and Hartshorn (1989) are the relative location of m achines, machining cost, processing time and availability of a machine. Tabucano n et al. (1994) propose an approach to the design and development of an intellig ent DSS that is intended to help the selection process of alternative machines f or FMS. The process consists of a series of steps starting with an analysis of t he information and culminating in a conclusion—a selection from several available alternatives and verification of the selected alternative to solve the problem. The approach combines the AHP technique with the rule-based technique for creati ng expert systems (ESs). This approach determines the architecture of the comput er-based

268 A. Bhattacharya et al. environment necessary for the decision support software system to be created. It includes the AHP software package (Expert Choice), Dbase III + DBMS, Expert Sys tem shell (EXSYS), and Turbo Pascal compiler (for the external procedural progra ms). A prototype DSS for a fixed domain, namely a CNC turning center that is req uired to process a family of rotational parts, is developed. Tabucanon et al.’s (1 994) methodology helps the user to find the most “satisfactory” machine on the basis of several objective as well as subjective attributes. Flexible manufacturing c ells (FMCs) have been used as a tool to implement flexible manufacturing process es to increase the competitiveness of manufacturing systems (Wang et al., 2000). In implementing an FMC, decision makers encounter the machine selection problem , including attributes, e.g., machine type, cost, number of machines, floor spac e, and planned expenditures (Wang et al., 2000). Wang et al. (2000) propose a fu zzy multiple-attribute decision-making (FMADM) model to assist the decision make r to deal with the machine selection problem for an FMC realistically and econom ically. In their work, the membership functions of weights for those attributes are determined in accordance with their distinguishability and robustness when t he ranking is performed. AHP is widely used for tackling FMS selection problems due to the concept’s simplicity and efficiency (Sambasivarao and Deshmukh, 1997). But AHP, as it is, do not take into consideration tangible factors, such as cost factors (Saaty, 1980, 1986, 1990). Thus, there is a need to allow cardinal fact ors in AHP to make the model robust and more efficient. In this chapter, a robus t MCDM procedure is proposed using AHP that incorporates qualitative as well as quantitative measures for the FMS selection problem. The methodology proposed is very useful first to quantify the intangible factors in a strong manner and the n to find out the best among member alternatives depending on their cost factors . Some researchers believe that qualitative benefits cannot be considered mathem atically unless one uses a knowledge-based system (Chen et al., 1998; Evans and Brown, 1989). This chapter outlines a fuzzified intelligent approach based on th e judgmental values of the decision maker in selecting the most cost-effective F MS. One objective of this chapter is to find out fuzziness patterns of FMS selec tion decisions having a disparate level-ofsatisfaction of the decision makers. A nother objective is to provide a robust, quantified monitor of the level of sati sfaction among decision makers and to calibrate these levels-of-satisfaction aga inst decision makers’ expectations.

FMS Selection Using a Fuzzy-MCDM Model 269 2. FMS SELECTION PROBLEM As a first step in testing the MCDM model proposed in the previous chapter, the authors have illustrated an example with FMS selection. Six different types of o bjective cost components have been identified for the selection problem. The tot al costs of each alternative are nothing, but the objective factor costs (OFCs) of the FMSs (refer to Table 1). The task is to select the best candidate-FMS amo ng five candidate-FMSs. Table 1. Cost Factor Components FMS/OFCs 1. Cost of Acquisition 2. Cost of Insta llation 3. Cost of Commissioning 4. Cost of Training 5. Cost of Operation 6. Cos t of Maintenance Total Cost (OFC) Objective Factor Measure (OFMi) S1 1.500 0.075 0.063 0.041 0.500 0.500 2.239 0.154 S2 0.800 0.061 0.052 0.043 0.405 0.405 1.43 1 0.241 S3 1.300 0.063 0.055 0.046 0.420 0.420 1.949 0.177 S4 1.00 0.053 0.050 0 .042 0.470 0.470 1.669 0.206 S5 0.900 0.067 0.061 0.040 0.430 0.430 1.550 0.222 The subjective attributes influencing the selection of FMS are shown in Table 2. The study consists of five different attributes, viz., flexibility in pick-up a nd delivery, flexibility in the conveying system, flexibility in automated stora ge and retrieval system, life expectancy/payback period, and tool magazine chang ing time. One may consider other attributes appropriate to selection of FMS. The attributes influencing the FMS selection problem are shown in Table 2. Table 2. Attributes Influencing the FMS Selection Problem Factor I Flexibility i n pick-up and delivery Factor II Flexibility in conveying system Factor III Flex ibility in automated storage and retrieval system Factor IV Life expectancy/pay back period Factor V Tool magazine changing time The MATLAB® fuzzy toolbox has been used in this work wherein a logical intelligent rule has been coded in M-file suitably using the designed MF.

270 A. Bhattacharya et al. 3. SIMULATION USING MATLB® The most important task for a decision maker is the selection of the factors. Th orough representation of the problem indicating the overall goal, criteria, subcriteria (if any), and alternatives in all levels maintaining the sensitivity to change in the elements is a vital issue. The number of criteria or alternatives in the proposed methodology should be reasonably small to allow consistent pair -wise comparisons. Matrix 1 is the decision matrix based on the judgmental value s from different judges. Matrices 2 to 6 show comparisons of the weightages for each attribute. Matrix 7 consolidates the results of the earlier tables in arriv ing at the composite weights, i.e., SFMi values, of each of the alternatives. 1 1 5 1 3 1 4 1 5 5 1 3 2 1 3 1 3 1 1 3 1 5 4 1 2 3 1 1 3 5 1 5 3 1 D = Matrix 1. Decision matrix (I.R. = 4.39%) 1 1 3 1 2 1 5 1 4 3 1 3 1 5 1 2 2 1 3 1 1 4 1 3 5 5 4 1 3 4 2 3 1 3 1 A1 = Matrix 2. Pair-wise comparison matrix for ‘F1’ (I.R. = 4.48%) 1 1 7 1 3 1 5 1 6 7 1 4 3 2 3 1 4 1 1 3 1 4 5 1 3 3 1 1 2 6 1 2 4 2 1 A2 = Matrix 3. Pair-wise comparison matrix

FMS Selection Using a Fuzzy-MCDM Model 1 1 4 A3 = 1 1 3 1 7 4 1 4 2 1 5 1 1 4 1 1 2 1 7 3 1 2 2 1 1 3 7 5 7 3 1 271 Matrix 4. Pair-wise comparison matrix F3 (I.R. = 1.88%) 1 3 A 4 for F2 (I.R. = 3.32%) for 6 6 3 2 1 1 3 1 1 5 1 7 1 6 5 5 1 1 2 1 3 3 7 2 1 1 2 = 1 5 1 3 1 6 Matrix 5. Pair-wise comparison matrix 1 3 A5 = 1 5 1 7 1 4 1 3 1 1 5 1 6 1 4 5 5 1 1 2 2 7 6 2 1 3 4 4 1 2 1 3 1 Matrix 6. Pair-wise comparison matrix for F4 (I.R. = 6.22%) and for F5 (I.R. = 6 .87%) 0.471 0.408 G = 0.159 0.279 0.050 0.103 0.076 0.512 0.051 0.246 0.117 0.075 0.25 9 0.366 0.104 0.338 0.151 0.040 0.131 0.273 0.501 0.103 0.075 0.047 0.063 0.305 0.458 0.074 0.047 0.116 Matrix 7. Final matrix to find out Global Priority In the proposed methodology, the unit of OFC is US$, whereas the objective factor measure (OFM) is a non dime nsional quantity. Correspon-

272 A. Bhattacharya et al. dingly, the SI is also a non-dimensional quantity. The higher the SI values, the better would be the selection. The value of the objective factor decision weigh t ( ) lies between 0 and 1. For = 0, SI = SFM; i.e., selection is solely depende nt on subjective factor measure values found from AHP and SFM values dominate ov er OFM values. There is no significance of considering the cost factor component s for = 0. For = 1, SI = OFM; i.e., OFM values dominate over the SFM values, and the FMS selection is dependent on OFM values only. For = 1, the cost factors ge t priority over the other factors. Keeping this in mind, the values of are taken in between 0 and 1. To verify the practicality and effectiveness of the final o utcome of the proposed methodology, sensitivity analysis is done. The basic fuzz ified equation governing the selection process is recalled once again. It is to be remembered that the Eq. (1) (Wabalickis, 1988) uses MF as depicted by Eq. (2) . LSIi SFMi LSI L 1 0.999 B 1 Ce 0.001 0 LSIU LSI L x x x ln a 1 C A LSIi 1 (1) xa x xb xb xb (2) x x xa x x The intelligent decision algorithm generates the coefficients of the fuzzy const raints in the decision variables. The rule first declares a function Cj and assi gns the constants in the MF. The aim is to produce a rule that works well on pre viously unseen data, i.e., the decision rule should “generalize” well. An example is appended below: function [cj] = mpgen(cj0,cj1,gamma,mucj) B = (0.998 / ((0.001 × exp(gamma)) 0.999)); A=0.999 × (1 + B); cj=cj0 + ((cj1 cj0) / gamma) × (log((1 / B) × ((A / mucj) 1)));

The rule supports this work by allowing the call to the function to contain a va riable, which is automatically set to different values as one may request. The l ogical way in which the intelligent fuzzy-MCDM acts as an agent in the entire sy stem includes many if else rules.

FMS Selection Using a Fuzzy-MCDM Model 273 3.1 Fuzzy Sensitivity of the MCDM Model In a real-life situation, the decision environments rarely remain static. Theref ore, it is essential to equip the proposed decision-making model with the capabi lity to determine changes in the selection process that results from making chan ges in the parameters of the model. So, the dynamic behavior of the optimal sele ction found from the proposed methodology can be checked through the fuzzy-sensi tivity plots. Among all the FMSs, FMS1 has the highest SI value when the objecti ve factor decision weight lies between 0.33 and 1.00. However, FMS2 would be pre ferred to other FMS candidate-alternatives when the value of level-ofsatisfactio n lies between 0.00 and 0.33. The appropriate value of the level-of-satisfaction is to be selected cautiously. The reason behind this is as follows. The higher the value, the dominance of the SFMi values will be higher. The lower the value, more will be the dominance of cost factor components, and subsequently, the int angible factors will get less priority. Table 3 illustrates the final ranking ba sed on the proposed model. From the Table 3 and Figures 16 to 20 ranking of the candidate-alternatives is FMS1 FMS2 FMS3 FMS5 FMS4, i.e., FMS1 is the best alter native at decision maker’s level-of-satisfaction = 0.42. Table 3 is a clear indica tion of accepting the proposed methodology for the selection problem in a confli cting-criteria environment. Relationship between the degree of fuzziness, , vers us level-ofsatisfaction ( ) has been depicted for all candidate-FMSs by Figures 1 to 5. This is a clear indication that the decision variables allow the MCDM mo del to achieve a higher level-of-satisfaction with a lesser degree of fuzziness. Figures 6 to 10 and 11 to 15 delineate SI indices versus level-of-satisfaction ( ) and SI indices versus degree of fuzziness ( ), respectively. 35 35 30 30 25 25 20 20 15 15 10 10 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 1. Fuzziness ( ) vs. for FMS1

contour plot Figure 2. Fuzziness ( ) vs. for FMS2 contour plot

274 35 A. Bhattacharya et al. 35 30 30 25 25 20 20 15 15 10 10 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 3. Fuzziness ( ) vs. for FMS3 35 contour plot Figure 4. Fuzziness ( ) vs. for FMS4 contour plot 0.36 30 0.34 0.32 25 0.3 20 0.28 0.26 15 0.24 10 0.22 0.2 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.18 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 5. Fuzziness ( ) vs. for FMS5 0.235 0.23 0.225 contour plot Figure 6. SI vs. contour plot for FMS1 0.25 0.24 0.23 0.22 0.215 0.21 0.205 0.2 0.22 0.21 0.2 0.19 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5

0.6 0.7 0.8 0.9 Figure 7. SI vs. contour plot for FMS2 Figure 8. SI vs. contour plot for FMS3

FMS Selection Using a Fuzzy-MCDM Model 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 275 0.2 0.18 0.16 0.14 0.12 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 9. SI vs. 0.36 0.34 contour plot for FMS4 Figure 10. SI vs. contour plot for FMS5 0.235 0.23 0.32 0.225 0.3 0.28 0.26 0.24 0.22 0.2 0.18 5 10 15 20 25 30 35 0.22 0.215 0.21 0.205 0.2 5 10 15 20 25 30 35 Figure 11. SI vs. contour plot for FMS1 0.25 0.24 Figure 12. SI vs. contour plot for FMS2 0.19 0.18 0.17 0.23 0.22 0.16 0.15 0.14

0.21 0.2 0.19 5 10 15 20 25 30 35 0.13 0.12 0.11 0.1 5 10 15 20 25 30 35 Figure 13. SI vs. contour plot for FMS3 Figure 14. SI vs. contour plot for FMS4

276 A. Bhattacharya et al. Figure 15. SI vs. contour plot for FMS5 Figure 16. Fuzzy-sensitivity for FMS1 Figure 17. Fuzzy-sensitivity for FMS2 Figure 18. Fuzzy-sensitivity for FMS3 Figure 19. Fuzzy-sensitivity for FMS4 Figure 20. Fuzzy-sensitivity for FMS5

FMS Selection Using a Fuzzy-MCDM Model 277 Combining the plots as illustrated in Figures 1 15, one gets Figures 16 20. Thes e figures elucidate 3-D mesh and contour plots. Basically Figures 16 20 illustra te fuzzy-sensitivity indicating relationships among SI indices, and . Furthermor e, from these plots, it is seen that the decision variables, as defined in Eq. ( 1), allow the MCDM model to achieve a higher level-ofsatisfaction ( ) with a les ser degree of fuzziness ( ). Table 3. Ranking of the Systems Candidate-FMS FMS1 FMS2 FMS3 FMS4 FMS5 SIi 0.249 0.224 0.210 0.155 0.162 Rank # #1 #2 #3 #5 #4 According to Table 3, the best alternative is FMS1 with the selection index of 0 .249. The worst alternative is FMS4 with the selection index of 0.155. 4. GENERAL DISCUSSIONS AND CONCLUSION This chapter outlined an intelligent fuzzy-MCDM model for appropriate selection of an FMS in a conflicting criteria environment. The proposed method calculates the GP for functional, design factors and other important attributes by eigenvec tor method of pair-wise comparison. These GPs are used as SFMs in determining SI . In a real-life situation, the decision environments rarely remain static. So, the dynamic behavior of the optimal selection found from the proposed methodolog y has been checked through the fuzzy-sensitivity plots. Figures 16 20 teach an i nteresting phenomenon that is found in nature. At a lower level-of-satisfaction ( ), the chances of getting involved in a higher degree of fuzziness ( ) increas e. Therefore, a decision maker’s level-of-satisfaction should be at least moderate in order to avoid higher degree of fuzziness while making any kind of decision using the proposed MCDM model delineated in the previous chapter. The methodolog y proposed is very useful first in quantifying the intangible factors in a stron g manner and then in finding out the best among

278 A. Bhattacharya et al. the alternatives depending upon their cost factors. Contrary to the traditional way of selection using discounted cash flow (DCF), this methodology is a sound a lternative to apply under an unstructured environment. The fuzzysensitivity stre ngthens the validity of the proposed methodology. It verifies the practicability as well as the effectiveness of the proposed DSS method. It is not possible for an individual to consider all the factors related to FMS as follows: FMSs are a vailable in a wide range, Performance standards of the systems are not uniform, and Expression of capabilities and performance attributes among manufacturers ar e inconsistent and incommensurable. Thus, a decision-making expert system may he lp the decision maker in selecting the most cost-effective FMS considering the c onflicting-in-nature factors of the systems. The selection problem of FMS is com plex due to the high capital costs involved and to the presence of multiple conf licting criteria. One can reduce investment and maintenance costs, increase equi pment utilization, increase efficiency, as well as improve facilities layout by selecting the right system suitable for the operations to be carried out. REFERENCES Abdel-Malek, L., and Wolf, C., 1991, Evaluating flexibility of alternative FMS d esigns A comparative measure, International Journal of Production Economics, 23( 1–3): 3–10. Barad, M., and Sipper, D., 1988, Flexibility in manufacturing systems: d efinitions and petri net modeling, International Journal of Production Research, 26: 237–248. Browne, J., Dubois, D., Rathmill, K., Sethi, S.P., and Stecke, K.E., 1984, Classification of flexible manufacturing systems, FMS Magazine, 2: 114–117. Buffa, E.S., 1993, Modern Production/Operations Management, Wiley Eastern Limit ed, New Delhi. Buzacott, J.A., and Mandelbaum, M., 1985, Flexibility and product ivity in manufacturing systems, Proceedings of the Annual IIE Conference, Los An geles, CA, pp. 404–413. Chen, Y., Tseng M.M., and Yien, J., 1998, Economic view of CIM system architecture, Production Planning & Control, 9(3): 241–249. Elango, B. , and Meinhart, W.A., 1994, Selecting a flexible manufacturing system: a strateg ic approach. Long Range Planning, 27(3): 118–126.

FMS Selection Using a Fuzzy-MCDM Model 279 Evans, G.W., and Brown, P.A., 1989, A multi objective approach to the design of flexible manufacturing systems, in Proceedings of International Industrial Engin eering Conference on Manufacturing and Societies, pp. 301–305. Gupta, Y.P., and Go yal, S., 1989, Flexibility of manufacturing systems: concepts and measurements, European Journal of Operational Research, 43: 119–135. Gindy, N.N.Z., and Ratchev, S.M., 1998, Integrated framework for machining equipment in selection of CIM, I nternational Journal of Computer Integrated Manufacturing, 11(4): 311–325. Haddock , J., and Hartshorn, T.A., 1989, A decision support system for specific machine selection, Computers & Industrial Engineering, 16(2): 277–286. Kaighobadi, M., and Venkatesh, 1994, Flexible manufacturing systems: an overview, International Jou rnal of Operations and Production Management, 14(4): 26–49. Karsak, E.E., 2002, Di stance-based fuzzy MCDM approach for evaluating flexible manufacturing system al ternatives, International Journal of Production Research, 40(13): 3167–3181. Karsa k, E.E., and Tolga, E., 2001, Fuzzy multi-criteria decision-making procedure for evaluating advanced manufacturing system investments, International Journal of Production Economics, 69: 49–64. Lenz, J.E., 1988, Flexible Manufacturing, Benefit s For The Low-Inventory Factory, Marcel Dekker, Inc., New York. Meredith, J.R., and Suresh, N.C., 1986, Justification techniques for advanced manufacturing tech nologies, International Journal of Production Research, 24: 1043–1057. Miltenburg, G.J., and Krinsky, I., 1987, Evaluating flexible manufacturing systems, IEEE Tr ansactions, 19: 222–233. Nagarur, N., 1992, Some performance measures of flexible manufacturing systems, International Journal of Production Research, 30: 799–809. Nelson, C.A., 1986, A scoring model for flexible manufacturing systems project s election, European Journal of Operational Research, 24: 346–359. Rai, R., Kameshwa ran, S., and Tiwari, M.K., 2002, Machine-tool selection and operation allocation in FMS: solving a fuzzy goal-programming model using a genetic algorithm, Inter national Journal of Production Research, 40(3): 641–665. Saaty, T.L., 1980, The An alytical Hierarchy Process, McGraw-Hill, New York. Saaty, T.L., 1990, How to mak e a decision: the analytic hierarchy process, European Journal of Operational Re search, 48(1): 9–26. Saaty, T.L., 1986, Exploring optimization through hierarchies and ratio scales, SocioEconomic Planning Sciences, 20(6): 355–360. Sambasivarao, K.V., and Deshmukh, S.G., 1997, A decision support system for selection and just ification of advanced manufacturing technologies, Production Planning and Contro l, 8: 270–284. Sarkis, J., and Talluri, S., 1999, A decision model for evaluation of flexible manufacturing systems in the presence of both cardinal and ordinal f actors, International Journal of Production Research, 37(13): 2927–2938. Shang, J. , and Sueyoshi, T., 1995, A unified framework for the selection of a flexible ma nufacturing system, European Journal of Operational Research, 85: 297–315.

280 A. Bhattacharya et al. Stam, A., and Kuula, M., 1991, Selecting a flexible manufacturing system using m ultiple criteria analysis, International Journal of Production Research, 29: 803–8 20. Tabucanon, M.T., Batanov, D.N., and Verma, D.K., 1994, Decision support syst em for multicriteria machine selection for flexible manufacturing systems, Compu ters in Industry, 25(2): 131–143. Trentesaux, D., Dindeleux, R., and Tahon, C., 19 98, A multicriteria decision support system for dynamic task allocation in a dis tributed production activity control structure. International Journal of Compute r Integrated Manufacturing, 11(1): 3–17. Vasant, P., Bhattacharya, A., and Barsoum , N. N., 2005, Fuzzy patterns in multi-level of satisfaction for MCDM model usin g smooth S-Curve MF, in:, Lecture Notes in Artificial Intelligence, Wang, L. and Jin, Y., (eds.), Springer-Verlag: Berlin, 3614: 1294–1303. Wabalickis, R.N., 1988 , Justification of FMS with the analytic hierarchy process, Journal of Manufactu ring Systems, 7: 175–182. Wang, T.Y., Shaw, C.F., and Chen, Y.-L., 2000, Machine s election in flexible manufacturing cell: a fuzzy multiple attribute decision-mak ing approach. International Journal of Production Research, 38(9): 2079–2097. Yurd akul, M., 2004, AHP as a strategic decision-making tool to justify machine tool selection, Journal of Materials Processing Technology, 146(3): 365–376.

SIMULATION SUPPORT TO GREY-RELATED ANALYSIS: DATA MINING SIMULATION David L. Olson1 and Desheng Wu2,3 1 Department of Management, University of Nebraska, Lincoln, NE 2Department of Mec hanical and Industrial Engineering, University of Toronto, Toronto, Ontario 3Sch ool of Business, University of Science and Technology of China, Hefei Anhui, Chi na Abstract: This chapter addresses the use of Monte Carlo simulation to reflect uncertainty as expressed by fuzzy input. Fuzziness is expressed through grey-related analysi s, using interval fuzzy numbers. The method standardizes inputs through norms of interval number vectors. Interval-valued indexes are used to apply multiplicati ve operations over interval numbers. The method is demonstrated on a practical p roblem. Simulation offers a more complete understanding of the possible outcomes of alternatives as expressed by fuzzy numbers. The focus is on probability rath er than on maximizing expected or extreme values. Fuzzy sets, Monte Carlo simula tion, grey-related analysis, data mining Key words: 1. INTRODUCTION This chapter addresses the use of Monte Carlo simulation to reflect uncertainty as expressed by fuzzy input. Simulation offers a more complete understanding of the possible outcomes of alternatives as expressed by fuzzy numbers. The focus i s on probability rather than on maximizing expected or extreme values. Both weig hts and alternative performance scores are allowed to be fuzzy. Both interval an d trapezoidal fuzzy input can be considered (see Olson and Wu, 2005, 2006). Fuzz y concepts have long been important in multiple criteria analysis (Dubois, 1980; Gau and Buehrer, 1993; Pawlak, 1982; Pearl, 1988; Pedrycz, 1998). Simulation ha s been applied to the analytical hierarchy C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 281

282 D.L. Olson and D. Wu process (AHP) (Levary and Wan, 1998), generating random pair-wise comparison inp ut values. The uncertainty and fuzziness inherent in decision making makes the u se of precise numbers problematic in multiattribute models. Decision makers are usually more comfortable providing intervals for specific model input parameters . Interval input in multiattribute decision making has been a very active field of research. Methods applying intervals have included (along with many others, s ee Zhang et al., 2005): 1. Use of interval numbers as the basis for ranking alte rnatives Brans and Vincke, 1985; El-Hawary, 1998; Chang and Yeh, 2004; Kahraman et al., 2004. 2. Error analysis with interval numbers Larichev and Moshkovich, 1 991. 3. Use of linear programming and object programming with feasible regions b ounded by interval numbers Roy, 1978; Liu et al., 1999; Royes et al., 2003. 4. U se of interval number ideal alternatives to rank alternatives by their nearness to the ideal Wang et al., 2004. AHP was presented (Saaty, 1977) as a way to take subjective human inputs in a hierarchy and to convert these to a value function . This method has proven extremely popular. Salo and Hamalainen (1992) published their interval method using linear programming over the constrained space of we ights and values as a means to incorporate uncertainty in decisionmaker inputs t o AHP hierarchies. The problem of synthesizing ratio judgments in groups was con sidered very early in AHP (Aczel and Saaty, 1983). Fuzzy AHP was proposed as ano ther way to reflect uncertainty in subjective inputs to AHP in the same group co ntext (Buckley, 1984; 1985a; 1985b). Simulation has been presented as a way to r ank order alternatives in the context of AHP values and weights (Levary and Wan, 1998). Other multiple criteria methods besides AHP have considered fuzzy input parameters. ELECTRE (Roy, 1978) and PROMETHEE (Brans and Vincke, 1985) have alwa ys allowed fuzzy input for weights. A multiattribute method involving fuzzy asse ssment for selection has been given

Simulation Support to Grey-Related Analysis 283 in the airline safety domain (Chang and Yeh, 2004) and for multiple criteria sel ection of employees (Royes et al., 2003). Sensitivity in multiattribute models w ith fuzzy inputs was considered by Aouam et al. (2003) and in goal programming b y Fan et al. (2004). Rough set applications have also been presented (Zaras, 200 4). This stream of research has obviously been rich and useful in application. I t is extended by grey-related analysis. 2. GREY-RELATED ANALYSIS Grey system theory was developed by Deng (1982) based on the concept that inform ation is sometimes incomplete or unknown. The intent is the same as with factor analysis, cluster analysis, and discriminant analysis, except that those methods often do not work well when sample size is small and sample distribution is unk nown (Wang et al., 2004). With greyrelated analysis, interval numbers are standa rdized through norms, which allow transformation of index values through product operations. The method is simple, practical, and demands less-precise informati on than other methods. Grey-related analysis and TOPSIS (Hwang and Yoon, 1981; L ai et al., 1994; Yoon and Hwang, 1995) both use the idea of minimizing a distanc e function. However, grey-related analysis reflects a form of fuzzification of i nputs and uses different calculations, to include a different calculation of nor ms. Feng and Wang (2001) applied grey relation analysis to select representative criteria among a large set of available choices and then used TOPSIS for outran king (Zhang et al., 2005) Grey-related analysis has been used in a number of app lications, In our discussion, we shall use the concept of the norm of an interva l number column vector, the distance between intervals, product operations, and number-product operations of interval numbers. Let a [a , a ] {x x a ,a a ,a ,a R} . a , we call We call a [a , a ] an interval number. If 0 a interval number a [a , a ] a posit ive interval number. Let X ([a1 , a1 ], [a 2 , a 2 ],..., [ a n , a n ]) T be an n -dimension interva l number column vector. a

284 D.L. Olson and D. Wu DEFINITION 1. If X ([a1 , a1 ], [a 2 , a 2 ],..., [a n , a n ]) T is an arbitrar y interval number column vector, the norm of X is defined here as X (1) DEFINITION 2. If a [a , a ] and b [b , b ] are two arbitrary interval numbers, t he distance from a [a , a ] to b [b , b ] , is defined as a b b , a b ) (2) DEFINITION 3. If k is an arbitrary positive real number, and a [a , a ] is an ar bitrary interval number, then k [a , a ] [ ka , ka ] will be called the number-p roduct between k and a [a , a ] . DEFINITION 4. If a [a , a ] is an arbitrary in terval number, and b [b , b ] are arbitrary interval numbers, we shall define th e interval number product [a , a ] [b , b ] as follows: when b when b 0 [a , a ] [b , b ] [a b , a b ] (3) (4) 0 [a , a ] [b , b ] [a b , a b ] If b+ = 0, the interval reverts to a point, and thus, we would return to the bas ic crisp model. 2.1 Steps of Grey-Related Analysis The principle and steps of the Grey-related analysis method are as follows: Step 1. Construct decision matrix A with an index number of interval numbers. If the index value of the jth index G j of feasible plan X i is an interval number [ai j , aij ] , i 1,2,..., m , j 1,2,..., n , decision matrix A with index number of interval numbers is defined as the follows: max( a max(max( a1 , a1 ), max( a2 , a2 ),..., max( an , an ))

Simulation Support to Grey-Related Analysis 285 [a11 , a11 ] [a12 , a12 ] ... [a1n , a1n ] A [a21 , a21 ] [a22 , a22 ] ... [a2 n , a2 n ] ... ... ... ... [am1 , am1 ] [am 2 , am 2 ] ... [amn , amn ] Step 2. Transform the “contrary index” into a positive index .The index is called a positive index if a greater index value is better. The index is called a contrar y index if a smaller index value is better. We may transform a contrary index in to a positive index if the jth index G j is a contrary index (5) [bij , bij ] [ aij , aij ] i 1,2,..., m . (6) Without loss of generality, in the following discussion, we supposed that all th e indexes are “positive indices.” Step 3. Standardize decision matrix A with an inde x number of interval numbers, obtaining standardizing decision matrix R [ rij ,r ij ] . If we mark the column vectors of decision matrix A with interval-valued i ndexes with A1 , A2 ,..., An , the element of standardizing decision matrix R [ rij ,rij ] is defined as [rij , rij ] [aij , aij ] i 1,2,..., m j 1,2,..., n . (7) Step 4. Calculate interval number weighted matrix C ([ cij ,cij ]) m n . The for mula for the element of interval number weighted matrix C is C ([ cij ,cij ]) m n where [cij , cij ] [c j , d j ] [rij , rij ] i 1,2,..., m j 1,2,..., n . (8) Step 5. Determine reference number sequence. The element of reference number seq uence is composed of the optimal weighted interval number index value for every alternative. A j

286 D.L. Olson and D. Wu U0 ([u 0 (1) , u 0 (1)] , [u 0 (2) , u 0 (2)] , ... , [u 0 (n) , u 0 (n)]) is a reference number sequence if u0 ( j ) max cij , u0 ( j ) 1 i m max cij , 1 i m j 1, 2,..., n . Step 6. Calculate connections between alternatives. First, calculate the connect ion coefficient i ( k ) between the sequence composed of weight interval number standardized index values for every alternative U i ([ci1 , ci1 ], [ci 2 , ci 2 ], ... , [cin , cin ] ) and the reference number sequence U 0 ([u0 (1), u0 (1)], [u0 (2), u0 (2)], ..., [u0 (n), u0 (n)]). The formula for i (k ) is (k ) min min [u0 (k ), u0 (k )] [cik , cik ] i k max max i k i k (9) i [u0 (k ), u0 (k )] [cik , cik ] max max [u0 (k ), u0 (k )] [cik , cik ] [u0 (k ), u0 (k )] [cik , cik ]

(0, ) , and is a resolving coefficient. The smaller is, Here [0, 1] .The value o f the greater its resolving power. In general, may be changed to reflect the des ired degree of resolution. After calculating i (k ) , the connection between the i-th plan and the reference number sequence is calculated by the following form ula: ri 1 n n i k 1 k , i 1, 2,...m (10) Step 7. Determine optimal plan. The feasible plan X t is optimal if rt max ri . 1 i m 3. MONTE CARLO SIMULATION

Fuzzy inputs can easily be simulated using Monte Carlo simulation models. Interv al random numbers over the interval 0 1 can be generated in Monte Carlo simulati on directly, and these can be converted to any other uniform range. Simulations can be easier to analyze if they are controlled, using unique seed values to ens ure that the difference in simulation output due to random variation was the sam e for each alternative.

Simulation Support to Grey-Related Analysis 287 3.1 Trapezoidal Distributed Fuzzy Numbers The trapezoidal fuzzy input dataset can also be simulated. X is random number (0 < rn < 1). Definition of trapezoida1 is left 0 in Figure 1; a2 is left 1; a3 is right 1; and a4 is right 0. 1.0 0 a1 a2 a3 a4 Figure 1. A trapezoidal fuzzy number J is area of left triangle contingent calculation: K is area of rectangle L is a rea of right triangle Fuzzy sum = left triangle + rectangle + right triangle = 1 M is the area of the left triangle plus the rectangle (for calculation of X val ue) X is the random number drawn (which is the area) If X J: X a1 X a2 a1 J a4 a3 a2 a1 L (11) If J X J+K: X a2 X K J a3 a2 (12) If J+K X: X a4 1 X a4 a3 a4 a3 a2 a1 J L (13)

288 D.L. Olson and D. Wu Our calculation is based on drawing a random g on the left (a1) as 0, ending on the right tance on the X-axis. The simulation software each model 1000 times for each random number the number of times each alternative won. 3.2 Grey-Related Decision Tree Models Grey-related analysis is expected to provide improvement over crisp models by be tter reflecting the uncertainty inherent in many human analysts’ minds. Data minin g models based on such data are expected to be less accurate, but hopefully not by very much (Hu et al., 2003). However, grey-related model input would be expec ted to be more stable under conditions of uncertainty where the degree of change in input data increased. We applied decision tree analysis to a small set (1000 observations total) of credit card data. Originally, there was one output varia ble (whether or not the account defaulted, a binary variable with 1 representing default, 0 representing no default) and 65 available explanatory variables. The se variables were analyzed, and 26 were selected as representing ideas that migh t be important to predicting the outcome. The original data set was imbalanced, with 140 default cases and 860 not defaulting. Initial decision tree models were almost all degenerate, classifying all cases as not defaulting. When differenti al costs were applied, the reverse degenerate model was obtained (all cases pred icted to default). Therefore, a new dataset containing all 140 default cases and 160 randomly selected not default cases was generated, where 200 cases were ran domly selected as a training set, with the remaining 100 cases used as a test se t. The explanatory variables included five binary variables and one categorical variable, with the remaining 20 being continuous. To reflect fuzzy input, each v ariable (except for binary variables) was categorized into three categories base d on analysis of the data, using natural cutoff points to divide each variable i nto roughly equal groups. Decision tree models were generated using the data min ing software PolyAnalyst. That software allows setting minimum support level (th e number of cases necessary to retain a branch on the decision tree), and a slid er setting to optimistically or pessimistically split criteria. Lower support le vels allow more branches, as does the optimistic setting. Every time the model w as run, a different decision tree was able to be obtained. number reflecting the area (startin (a4) as 1), and calculating the dis Crystal Ball was used to replicate seed. The software enabled counting

Simulation Support to Grey-Related Analysis 289 But nine settings were applied, yielding many overlapping models. Three unique d ecision trees were obtained, which are reflected in the output to follow. A tota l of eight explanatory variables were used in these three decision trees. The sa me runs were made for the categorical data reflecting grey-related input. Four u nique decision trees were obtained, with formulas again given below. A total of seven explanatory variables were used in these four categorical decision trees. All seven models and their fit on test data are given in the Appendix. These mod els were then entered into a Monte Carlo simulation (supported by Crystal Ball s oftware). A perturbation of each input variable was generated, set at five diffe rent levels of perturbation. The intent was to measure the loss of accuracy for crisp and grey-related models. The model results are given in the seven model re ports in the appendix. Since different variables were included in different mode ls, it is not possible to directly compare relative accuracy as measured by fitt ing test data. However, the means for the accuracy on test data for each model g iven in Table 1 show that the crisp models declined in accuracy more than the ca tegorical models. The column headings in Table 1 reflect the degree of perturbat ion simulated. Table 1. Mean Model Accuracy Model Cont. 1 Cont. 2 Cont. 3 Cont. Cat. 1 Cat. 2 C at. 3 Cat. 4 Cat. Crisp 0.70 0.67 0.71 0.693 0.70 0.70 0.70 0.70 0.700 0.25 0.70 0.67 0.71 0.693 0.70 0.70 0.70 0.70 0.700 0.50 0.70 0.67 0.70 0.690 0.68 0.70 0 .70 0.70 0.695 1.00 0.68 0.67 0.69 0.680 0.67 0.69 0.69 0.69 0.688 2.00 0.67 0.6 7 0.67 0.670 0.66 0.68 0.69 0.68 0.678 3.00 0.66 0.66 0.67 0.667 0.66 0.67 0.68 0.67 0.670 4.00 0.65 0.66 0.66 0.657 0.65 0.67 0.67 0.67 0.665 The fuzzy models were expected to be less accurate, but here they actually avera ge slightly better accuracy. This, however, can simply be attributed to differen t variables being used in each model. The one exception is that models Continuou s 2 and Categorical 3 were based on one variable, V64, the balance-to-payment ra tio. The cutoff generated by model Continuous 2 was 6.44 (if V64 was < 6.44, pre diction 0), whereas the cutoff for Categorical 3 was 4.836 (if V64 was > 4.835, the category was “high,” and the decision tree model was that if V64 = “high,” predictio n 1, else prediction 0). The fuzzy model here was actually better in fitting the test data (although slightly worse in fitting the training data).

290 D.L. Olson and D. Wu The important point of the numbers in Table 1 is that there clearly was greater degradation in model accuracy for the continuous models than for the categorical (grey-related) models. This point is demonstrated further by the wider dispersi on of the graphs in the Appendix. 4. CONCLUSIONS This chapter has discussed the integration of grey-related analysis and decision making with uncertainty through simulation. Simulation provides a means to bett er visualize model results and a flexible way to include any level of uncertaint y and complexity. Results based on Monte Carlo simulation as a data-mining techn ique offer more insights to assist our decision making in fuzzy environments by incorporating probability interpretation. Analysis of decision tree models throu gh simulation shows that there does appear to be less degradation in model fit f or grey-related (categorical) data than for decision tree models generated from raw continuous data. It must be admitted that this is a preliminary result, base d on a relatively small dataset of only one type of data. However, it is intende d to demonstrate a point meriting future research. This decisionmaking approach can be applied to large-scale datasets, expanding our ability to implement data mining and large-scale computing. The easiest way to apply fuzzy concepts to dat a mining is to categorize data. This creates the problem of where to set limits between categories. However, reliance on expert judgment can often provide usefu l limits. If data-mining data are represented through fuzzy concepts, simulation can be applied. Since fuzzy data are probabilistic, simulation seems appropriat e. Simulation does involve a lot more work than closed-form (crisp) datasets. Ho wever, fuzzy data are often a better representation of real domains. REFERENCES Aczel, J., and Saaty, T.L., 1983, Procedures for synthesizing ratio judgments. J ournal of Mathematical Psychology, 27: 93 102. Aouam, T., Chang, S.I., and Lee, E.S., 2003, Fuzzy MADM: An outranking method, European Journal of Operational Re search, 145(2): 317–328. Brans, J.P., and Vincke, Ph., 1985, A preference ranking organization method: The PROMETHEE method. Management Science, 31: 647–656. Buckle y, J.J., 1984, The multiple judge, multiple criteria ranking problem: A fuzzy se t approach. Fuzzy Sets and System, 13(1): 25–37.

Simulation Support to Grey-Related Analysis 291 Buckley, J.J., 1985a, Ranking alternatives using fuzzy members. Fuzzy Sets and S ystems, 17: 233 247. Buckley, J.J., 1985b, Fuzzy hierarchical analysis. Fuzzy Se ts and Systems, 17: 233 247. Chang, Y.-H., and Yeh, C.-H., 2004, A new airline s afety index, Transportation Research Part B, 38: 369 383. Deng, J.L., 1982, Cont rol problems of grey systems. Systems and Controls Letters, 5: 288 294. Dubois, D., and Prade, H., 1980, Fuzzy Sets and Systems: Theory and Applications, Academ ic Press, Inc., New York. El-Hawary, M.E., 1998, Electric Power Applications of Fuzzy Systems, The Institute of Electrical and Electronics Engineers Press, Inc. , New York. Fan, Z., Hu, G., and Xiao, S.-H., 2004, A method for multiple attrib ute decision-making with the fuzzy preference relation on alternatives, Computer s & Industrial Engineering, 46: 321 327. Feng, C.-M., and Wang, R.-T., 2001, Con sidering the financial ratios on the performance evaluation of highway bus indus try, Transport Reviews, 21(4): 449 467. Gau, W.L., and Buehrer, D.J., 1993, Vagu e sets, IEEE Transactions On Systems, Man, And Cybernetics, 23: 610 614. Hu, Y., Chen, R.-S., and Tzeng, G.-H., 2003, Finding fuzzy classification rules using d ata mining techniques, Pattern Recognition Letters, 24(1 3): 509 519. Hwang, C.L ., and Yoon, K., 1981, Multiple Attribute Decision Making: Methods and Applicati ons, Springer-Verlag, New York. Kahraman, C., Cebeci, U., and Ruan, D., 2004, Mu lti-attribute comparison of catering service companies using fuzzy AHP: the case of Turkey, International Journal of Production Economics, 87: 171 184. Lai, Y.J., Liu, T.-Y., and Hwang, C.-L., 1994, TOPSIS for MODM, European Journal of Ope rational Research, 76(3): 486 500. Larichev, O.I., and Moshkovich, H.M., 1991, Z APROS: A Method And System For Ordering Multiattribute Alternatives On The Base Of A Decision-Maker’s Preferences, All-Union Research Institute for System Studies , Moscow. Levary, R.R., and Wan, K., 1998, A simulation approach for handling un certainty in the analytic hierarchy process, European Journal of Operational Res earch, 106: 116 122. Liu, S., Guo, B., and Dang, Y., 1999, Grey System Theory an d Applications, Scientific Press, Beijing. Olson, D.L., and Wu, D., 2005, Decisi on making with uncertainty and data mining, The 1st International Conference on Advanced Data Mining and Applications (ADMA2005), Li, X., Wang, S., and Yang D. Z., eds., Lecture Notes in Computer Science, Springer, Berlin. Olson, D.L., and Wu, D., 2006, Simulation of fuzzy multiattribute models for grey relationships, European Journal of Operational Research, 175(1): 111 120. Pawlak, Z., 1982, Rou gh sets, International Journal of Information & Computer Sciences, 11: 341 356. Pearl, J., 1988, Probabilistic Reasoning in Intelligent Systems: Networks of Pla usible Inference, Morgan Kaufmann, San Mateo, CA. Pedrycz, W., 1998, Fuzzy set t echnology in knowledge discovery, Fuzzy Sets and Systems, 98(3): 279 290. Rocco S., and Claudio, M., 2003, A rule induction approach to improve Monte Carlo syst em reliability assessment, Reliability Engineering and System Safety, 82(1): 85 92.

292 D.L. Olson and D. Wu Roy, B., 1978, ELECTRE III: un algorithme de classement fonde sur une representa tion floue des preferences en presence de criteres multiple, Cahiers du Centre E tudes Recherche Operationelle, 20: 3 24. Royes, G.F., Bastos, R.C., and Royes, G .F, 2003, Applicants’ selection applying a fuzzy multicriteria CBR methodology, Jo urnal of Intelligent & Fuzzy Systems, 14(4): 167 180. Saaty, T.L., 1977, A scali ng method for priorities in hierarchical structures, Journal of Mathematical Psy chology, 15: 234 281. Salo, A.A., and Hamalainen, R.P., 1992, Preference assessm ent by imprecise ratio statements, Operations Research, 40: 1053 1061. Wang, R.T., Ho, C.-T., Feng, C.-M., and Yang, Y.-K., 2004, A comparative analysis of the operational performance of Taiwan’s major airports, Journal of Air Transport Mana gement, 10: 353 360. Yoon, K., and Hwang, C.L., 1995, Multiple Attribute Decisio n Making: An Introduction Sage, Thousand Oaks, CA. Zaras, K., 2004, Rough approx imation of a preference relation by a multi-attribute dominance for deterministi c, stochastic and fuzzy decision problems, European Journal of Operational Resea rch, 159: 196 206. Zhang, J., Wu, D., and Olson, D.L., 2005, The method of grey related analysis to multiple attribute decision making problems with interval nu mbers, Mathematical and Computer Modelling, 42(9–10): 991–998.

Simulation Support to Grey-Related Analysis 293 APPENDIX: MODELS AND THEIR RESULTS Continuous Model 1: IF(Bal/Pay<6.44,N,IF(Utilization<1.54,Y,IF(AvgPay<3.91,N,Y)) ) Forecast: Cont M1 accuracy 1,000 Trials .374 .281 .187 .094 .000 0.68 0.69 0.70 proportion 0.72 0.73 Frequency Chart 994 Displayed 374 280.5 187 93.5 0 Test matrix: Model 0 Actual 0 43 Actual 1 14 Model 1 16 27 Accuracy 0.70 Simulation accuracy of 100 observations, 1000 simulation runs perturbation [ 0.2 5,0.25] 0.67 0.73 perturbation [ 0.50,0.50] 0.65 0.74 perturbation [ 1,1] 0.62 0 .75 perturbation [ 2,2] 0.58 0.74 perturbation [ 3,3] 0.57 0.74 .55 .6 .65 .7 .75

294 Continuous Model 2: IF (Bal/Pay<6.44,N,Y) D.L. Olson and D. Wu Forecast: Cont M2 accuracy 1,000 Trials .466 .350 .233 .117 .000 0.65 0.66 0.67 proportion 0.69 0.70 Frequency Chart 991 Displayed 466 349.5 233 116.5 0 Test matrix: Model 0 Actual 0 40 Actual 1 14 Model 1 19 27 Accuracy 0.67 Simulation accuracy of 100 observations, 1000 simulation runs perturbation [ 0.2 5,0.25] 0.65 0.71 perturbation [ 0.50,0.50] 0.63 0.71 perturbation [ 1,1] 0.60 0 .74 perturbation [ 2,2] 0.58 0.75 perturbation [ 3,3] 0.55 0.78 .55 .6 .65 .7 .75

Simulation Support to Grey-Related Analysis Continuous Model 3: IF(Bal/Pay<6.44,N,IF(Utilization<1.54,Y,IF(AvgRevPay<2.28,Y, N))) 295 Forecast: Cont M3 accuracy 1,000 Trials .237 .178 .119 .059 .000 Frequency Chart 996 Displayed 237 177.7 118.5 59.25 0 0.65 0.68 0.70 proportion 0.73 0.75 Test matrix: Model 0 Actual 0 44 Actual 1 14 Model 1 15 27 Accuracy 0.71 Simulation accuracy of 100 observations, 1000 simulation runs perturbation [ 0.2 5,0.25] 0.65 0.76 perturbation [ 0.50,0.50] 0.63 0.76 perturbation [ 1,1] 0.59 0 .77 perturbation [ 2,2] 0.54 0.79 perturbation [ 3,3] 0.53 0.78 .55 .6 .65 .7 .75

296 Categorical Model 1: D.L. Olson and D. Wu IF(Bal/Pay<6.44,N,IF(Utilization<1.54,Y,IF(AvgRevPay<2.28,Y,N))) Forecast: CatM1 accuracy 1,000 Trials .416 .312 .208 .104 .000 0.66 0.67 0.68 proportion 0.69 0.70 Frequency Chart 999 Displayed 416 312 208 104 0 Test matrix: Model 0 Actual 0 33 Actual 1 5 Model 1 26 36 Accuracy 0.70 Simulation accuracy of 100 observations, 1000 simulation runs perturbation [ 0.2 5,0.25] 0.66 0.71 perturbation [ 0.50,0.50] 0.64 0.71 perturbation [ 1,1] 0.61 0 .71 perturbation [ 2,2] 0.58 0.73 perturbation [ 3,3] 0.56 0.74 .55 .6 .65 .7 .75

Simulation Support to Grey-Related Analysis Categorical Model 2: IF(Bal/Pay= ”high”,IF(CredLine=”low”, IF(CDL=”mid”,IF(Pur%Bal=”low”,Y, F(CDL=”low”,N,Y)) IF(CredLine=”high”,IF(CalcIntRate=”mid”,N,Y),Y),N) 297 Forecast: CatM2 accuracy 1,000 Trials .222 .167 .111 .056 .000 0.65 0.68 0.70 proportion 0.73 0.75 Frequency Chart 997 Displayed 222 166.5 111 55.5 0 Test matrix: Model 0 Actual 0 42 Actual 1 13 Model 1 17 28 Accuracy 0.70 Simulation accuracy of 100 observations, 1000 simulation runs perturbation [ 0.2 5,0.25] 0.65 0.75 perturbation [ 0.50,0.50] 0.64 0.76 perturbation [ 1,1] 0.61 0 .76 perturbation [ 2,2] 0.58 0.76 perturbation [ 3,3] 0.57 0.80 .55 .6 .65 .7 .75

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298 Categorical Model 3: IF(Bal/Pay=”high”,Y,N) D.L. Olson and D. Wu Forecast: CatM3 accuracy 1,000 Trials .680 .510 .340 .170 .000 Frequency Chart 982 Displayed 680 510 340 170 0 0.68 0.69 0.69 proportion 0.70 0.70 Test matrix: Actual 0 Actual 1 Model 0 33 4 Model 1 26 37 Accuracy 0.70 Simulation accuracy of 100 observations, 1000 simulation runs perturbation [ 0.2 5,0.25] 0.68 0.70 perturbation [ 0.50,0.50] 0.67 0.71 perturbation [ 1,1] 0.66 0 .72 perturbation [ 2,2] 0.62 0.73 perturbation [ 3,3] 0.59 0.75 .55 .6 .65 .7 .75

Simulation Support to Grey-Related Analysis Categorical Model 4: IF(Bal/Pay=”high”, IF(CredLine=”low”, IF(CDL=”mid”,IF(Purch%Bal=”low”, IF(CDL=”low”,IF(Residence<.5,Y,N),Y)) IF(CredLine=”high”,IF(CalcIntRate=”mid”,N,Y),Y) 299 Forecast: CatM4 accuracy 1,000 Trials .236 .177 .118 .059 .000 0.66 0.69 0.71 proportion 0.74 0.76 Frequency Chart 998 Displayed 236 177 118 59 0 Test matrix: Actual 0 Actual 1 Model 0 41 12 Model 1 18 29 Accuracy 0.70 Simulation accuracy of 100 observations, 1000 simulation runs perturbation [ 0.2 5,0.25] 0.65 0.76 perturbation [ 0.50,0.50] 0.64 0.77 perturbation [ 1,1] 0.61 0 .77 perturbation [ 2,2] 0.58 0.77 perturbation [ 3,3] 0.57 0.77 .55 .6 .65 .7 .75

NEURO-FUZZY APPROXIMATION OF MULTI-CRITERIA DECISION-MAKING QFD METHODOLOGY Ajith Abraham1, Pandian Vasant2, and Arijit Bhattacharya3 1 Center of Excellence for Quantifiable Quality of Service, Norwegian University o f Science and Technology, Trondheim, Norway 2EEE Program Research Lecturer Unive rsiti Teknologi Petronas, Perak DR, Malaysia 3Embark Initiative Post-Doctoral Re search Fellow, School of Mechanical & Manufacturing Engineering, Dublin City Uni versity, Glasnevin, Dublin 9, Ireland This chapter demonstrates how a neuro-fuzz y approach could produce outputs of a further-modified multi-criteria decision-m aking (MCDM) quality function deployment (QFD) model within the required error r ate. The improved fuzzified MCDM model uses the modified S-curve membership func tion (MF) as stated in an earlier chapter. The smooth and flexible logistic memb ership function (MF) finds out fuzziness patterns in disparate level-of-satisfac tion for the integrated analytic hierarchy process (AHP-QFD model. The key objec tive of this chapter is to guide decision makers in finding out the best candida te-alternative robot with a higher degree of satisfaction and with a lesser degr ee of fuzziness. ANFIS, AHP, QFD, fuzziness patterns, decision-making, level-ofsatisfaction Abstract: Key words: 1. INTRODUCTION Arriving at the decision to install a robot in a manufacturing firm can be a dif ficult and complicated process. Even after the initial decision to acquire a rob ot is made, the problem of which robot to select from the many that are availabl e can confound managers who often lack the time and expertise to perform an exte nsive search and analysis. Furthermore, the current trend indicates that the num ber of robot manufacturers and suppliers are increasing as engineers continue to find more applications for robots. The C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 301

302 A. Abraham et al. problem of robot selection has become more difficult in recent years due to incr easing complexity, available features, and facilities offered by different robot ic products. 1.1 Concepts on Neuro-Fuzzy Systems A fuzzy inference system (FIS) can use human expertise by storing its essential components in the rule base and the database and can perform fuzzy reasoning to infer the overall output value. The derivation of if then rules and correspondin g membership functions (MFs) depends heavily on the a priori knowledge about the system under consideration. However, there is no systematic way to transform ex periences of knowledge of human experts into the knowledge base of an FIS. There is also a need for adaptability or some learning algorithms to produce outputs within the required error rate. On the other hand, ANN learning mechanism does n ot rely on human expertise. Due to the homogenous structure of ANN, it is hard t o extract structured knowledge from either the weights or the configuration of t he an artificial neural network (ANN). The weights of the ANN represent the coef ficients of the hyperplane that partition the input space into two regions with different output values. If we can visualize this hyperplane structure from the training data, then the subsequent learning procedures in an ANN can be reduced. However, in reality, the a priori knowledge is usually obtained from human expe rts; it is most appropriate to express the knowledge as a set of fuzzy if then r ules, and it is not possible to encode into an ANN 0. Table 1 summarizes the com parison of FIS and ANN. Table 1. Complementary Features of ANN and FIS ANN Black box Learning from scrat ch FIS Interpretable Making use of linguistic knowledge To a large extent, the drawbacks pertaining to these two approaches seem complem entary. Therefore it is natural to consider building an integrated system combin ing the concepts of FIS and ANN modeling. A common way to apply a learning algor ithm to a FIS is to represent it in a special ANN like architecture 0. However, the conventional ANN learning algorithms (gradient descent) cannot be applied di rectly to such a system as the functions used in the inference process are usual ly nondifferentiable. This problem can be tackled by using differentiable

Neuro-Fuzzy Approximation of MCDM QFD 303 functions in the inference system or by not using the standard neural learning a lgorithm. In our simulation, we used an adaptive network based fuzzy inference s ystem (ANFIS) (Jang, 1991). ANFIS implements a Takagi Sugeno Kang (TSK) fuzzy in ference system (Jang, 1991) in which the conclusion of a fuzzy rule is constitut ed by a weighted linear combination of the crisp inputs rather than by a fuzzy s et. For a first-order TSK model, a common rule set with two fuzzy if then rules is represented as follows: Rule 1: If x is A1 and y is B1, then f1 = p1x + q1y + r1 Rule 2: If x is A2 and y is B2, then f2 = p2x + q2y + r2 where x and y are l inguistic variables and A1, A2, B1, and B2 are corresponding fuzzy sets and p1, q1, r1 and p2, q2, r2 are linear parameters. Figure 1. TSK type fuzzy inference system Figure 1 illustrates the TSK fuzzy inference system when two membership function s each are assigned to the two inputs (x and y). The TSK fuzzy controller usuall y needs a smaller number of rules, because their output is already a linear func tion of the inputs rather than a constant fuzzy set. Figure 2 depicts the five-l ayered architecture of ANFIS, and the functionality of each layer is as follows: Layer-1. Every node in this layer has a node function Oi1 1 Oi Ai ( x ) , for i = 1, or 2 ( y) , Bi 2 for i = 3,4,…. is the membership grade of a fuzzy set A ( = A1, A2, B1 or B2), and it specifies the degree to which the given input x (or y) satisfies the quantifier A. Usuall y the node function can be any parameterized function. Oi1

304 A. Abraham et al. A Gaussian membership function is specified by two parameters c (membership func tion center) and (membership function width). 2. ADAPTIVE NETWORK-BASED FUZZY INFERENCE SYSTEM (ANFIS) Figure 2. Architecture of the ANFIS 1 x c 2 2 Guassian (x, c, ) = e . Parameters in this layer are referred to as premise parameters. Layer-2. Every n ode in this layer multiplies the incoming signals and sends the product out. Eac h node output represents the firing strength of a rule. Oi2 wi Ai ( x ) Bi ( y ) ,i 1, 2 . In general any T-norm operators perform fuzzy AND can be used as the n ode function in this layer.

Neuro-Fuzzy Approximation of MCDM QFD 305 Layer-3. Every ith node in this layer calculates the ratio of the ith rule’s firin g strength to the sum of all rule’s firing strength. Oi3 wi wi w1 w2 ,i 1, 2 . Layer-4. Every node i in this layer is with a node function O14 wi f i wi ( p i x qi y ri ) , where wi is the output of layer-3, and p i , q i , ri is the parameter set. Para meters in this layer will be referred to as consequent parameters. Layer-5. The single node in this layer computes the overall output as the summation of all in coming signals: O15 Overall output i w if i i wi f i i wi ANFIS makes use of a mixture of backpropagation to learn the premise parameters and least mean square estimation to determine the consequent parameters. A step in the learning procedure has two parts: In the first part, the input patterns a re propagated, and the optimal conclusion parameters are estimated by an iterati ve least mean square procedure, whereas the antecedent parameters (membership fu nctions) are assumed to be fixed for the current cycle through the training set. In the second part, the patterns are propagated again, and in this epoch, backp ropagation is used to modify the antecedent parameters, whereas the conclusion p arameters remain fixed. This procedure is then iterated (Jang, 1991). 3.

QFD PROCESS QFD is a method for structured product planning and development. It enables a de velopment team to specify clearly the customer’s requirement. It also evaluates ea ch proposed product systematically in terms of its impact on meeting those requi rements (Hauser and Clausing, 1988; Wasserman, 1993). It is also an important to ol for concurrent engineering. In the era of globalization, the customer’s order d ecoupling point (CODP)

306 A. Abraham et al. is at make-to-order (MTO) stage (Bhattacharya et al., 2005). From Figure 1 it is understood where to apply the QFD process. QFD is used at a CODP to ensure that the voice of the customer is heard throughout the product planning and design s tage (Franceschini and Rosetto, 1995). QFD, in fact, is a method of continuous p roduct improvement, emphasizing the impact of organizational learning on innovat ion (Govers, 2001). Component Level Assembly Level Finished Goods Raw Material M A N U F A C T U R E R MTS ATO MTO ETO C U S T O M E R MTS: Make To Stock ATO: Assemble To Order MTO: Make To Order ETO: Engineer To Or der : Customer Order Decoupling Point Figure 3. Relationship between CODP and MCDM-QFD process (Bhattacharya et al., 2 005) In QFD process, a matrix called the house-of-quality (HOQ) (Hauser and Clausing, 1988) is used to display the relationship between the voice of customers (WHATs ) and the quality characteristics (HOWs) (Chuang, 2001). WHATs and HOWs are noth ing but the customer and technical requirements, respectively. The HOQ is develo ped during the QFD transformation. Basically the HOQ demonstrates how the techni cal requirements satisfy the customer requirements. The matrix highlights the im portant issues in the planning of a new product or improving an existing product . QFD, when combining WHATs and HOWs with competitive analysis (WHYs), represent s a customer-driven and market-oriented process for decision making (Cohen, 1995 ). A traditional QFD model uses absolute importance to identify the degree of im portance for each customer requirement. The psychology of customers, in general, is to rate almost everything as equally important,

Neuro-Fuzzy Approximation of MCDM QFD 307 although it is not. As the absolute weighing data tend to be bunched near the hi ghest possible scores, the differentiation of customer requirements is thus stro ngly recommended. These data, as they are, do not contribute much to helping QFD developers in prioritizing technical responses. At this juncture, the AHP (Saat y, 1988; 1990; 1994) prioritizes the customer’s requirements by putting the relati ve degree of importance to each customer-requirement. The task of the QFD team i s to list the technical requirements (TRs). These requirements are most likely t o affect the CRs. TR evaluators, in the QFD team, evaluate how the competitors’ pr oducts compare with that of company’s product. This evaluation leads to fixing of technical targets. From the QFD matrix, the discrepancies, if any, between the c ustomers’ perception and the QFD team’s correlation of CR and TR can be easily under stood. The vertical part of the QFD matrix shows how the company may respond to customer requirements. 4. DEVELOPMENT OF THE COMBINED AHP-QFD METHODOLOGY The methodology integrating the MCDM methodology (AHP) and QFD for a selection p roblem comprises the following steps and is shown in Figure 4: Step 1. Identific ation of customer requirements. Step 2. Identification of technical requirements . Step 3. Construction of central relationship matrix using expert knowledge of QFD team. Step 4. Computation of degree of importance for customer requirements by using AHP. Step 5. Computation of the degree of importance of technical requi rements by Eq. (1). m (1) wj = Rij ci i =1 importance degree of the jth technical requirement j 1, 2, ..., n , Rij = quanti fied relationship between the ith customer requirement and the jth technical cri teria in the central relationship matrix, and ci = importance weighing of the it h customer requirement. where wj =

308 Start A. Abraham et al. Identify customer requirements (CR) and technical requirements (TR) Generate central relationship matrix Change the judgmental values Computation of degree of importance for CR using AHP Calculate PV max, IR No Is I.R. < 10%? Yes Computation and Normalization of degree of importance for TR Change the judgmental Generate Pair-wise Comparison Matrices for each TR using AHP Calculate PV, max, IR No Is I.R. < 10%? Yes Evaluation of score for each TR for each candidate- alternative Computation of overall score (SFMi) for each candidate- alternative Compute OFM for each candidate- alternative Include modified S-Curve MF in SI equation Re - design S-Curve membership function for SI equation No Is degree of fuzziness > preferred value? Yes Maximize SIi values Modified fuzzy S-curve MF Rank the alternatives based on SIi values

End Figure 4. Flowchart of the proposed methodology

Neuro-Fuzzy Approximation of MCDM QFD 309 Step 6. Normalization of the degree of importance of technical criteria by Eq. ( 2). w j = w j ×100 n wj j 1 __ (2) Step 7. Construction of pair-wise comparison matrices for each technical require ment using Saaty’s (1988; 1990) nine-point scale. Step 8. Evaluation of score, wij , for each technical requirement for each candidate-alternative. Step 9. Comput ation of overall score (Chuang, 2001) by using Eq. (3). n __ Sj j 1 w j eij (3) where, S j = overall score for the jth candidate-alternative j 1, 2, ..., n , w j = normalized importance degree of the jth technical criteria j 1, 2, ..., n , and eij = PV value of the jth alternative on the ith technical criteria Step 10. Computation of OFM values for each candidate robot by using Eq. 4. OFM = Object ive Factor Measure, OFC = Objective Factor Cost, SFM = Subjective Factor Measure , SI = Selection Index, = Objective factor decision weight, and n = number of ca ndidate-alternatives (n = 4 in for the robot selection problem). OFMi = [ OFCi × ( OFCi–1 ) ]–1 (4) Step 11. Identification of fuzziness patterns and measurement of levelof- satisf action of the decision maker using modified S-curve MF. Step 12. Re designing th e MF if the degree of fuzziness is greater than a preferred value. Step 13. Maxi mization of the SI (selection index) value using Eq. 5. SIi = [ ( × SFMi ) + ( 1 ) × OFMi ] (5)

310 A. Abraham et al. Step 14. Ranking of all the candidate-alternatives Step 15. Selection of the bes t candidate-alternative using the analogy the higher the score, the better the s election. 5. ROBOT SELECTION PROBLEM An illustrative example of a process industry dealing with an enormous volume of manufactured product was illustrated by Bhattacharya et al. (2005). Out of four robots, the best-suited robot was purchased for the desired job for a very spec ific manufacturing process using the methodology of combined AHP-QFD as depicted by Bhattacharya et al. (2005). But what is lacking in the said proposed model o f Bhattacharya et al. (2005) is the evaluation of the fuzzy parameters in their multi-criteria selection model. When fuzzy parameters like human expertise and l inguistic knowledge get involved with the model, there is always a need for the model to approximate the outputs within the required error rate. Thus, the ANFIS (Jang, 1991) is found suitable in dealing with this complex problem of multi-cr iteria decision making. Considering the robot selection data of Bhattacharya et al. (2005) we begin with fitting the modified S-curve MF (Eq. 6) in their method ology. Step 11 onward of the methodology have been proposed herein with the fuzz y S-curve MF. 1 0.99 B 1 Ce yx 0.001 0 x x xa x x xa xa x xb xb xb (6) x We use the previously identified customer requirements (CRs) viz., payload, accu racy, life-expectancy, velocity, programming flexibility and total cost of robot , and seven TRs, viz., drive system, geometrical dexterity, path measuring syste m, size, material, weight and initial operating cost of robot. As in the case of Bhattacharya et al. (2005) the job is to select the best one of the four robots . The additional purposes of the current model are to view the fuzziness pattern s as well as the level-of-

Neuro-Fuzzy Approximation of MCDM QFD 311 satisfaction of the decision maker, and to approximate the model with a predeter mined allowable error rate. For measuring the relative degree of importance for each customer requirement, based on the proposed methodology, a (6 × 6) decision m atrix is constructed and shown in Figure 5. 1 7 1/ 7 1 1/ 3 3 3 4 5 1/ 3 1/ 2 2 1 3 6 9 3 2 4 1/ 7 1 1 7 3 1 4 3 5 6 3 9 3 2 4 0.143 1 = 0.143 1 0.333 3 0.250 2 0.333 0.500 2 0.333 1 D= 1/ 4 2 1/ 3 1 3 1/ 5 1/ 2 1/ 6 1/ 3 1 1/ 9 1/ 3 1/ 2 1/ 4 7 0.200 0.500 0.167 0.333 1 0.111 0.333 0.500 0.250 7 Figure 5. Decision matrix of this decision matrix are found and max , I.I., R.I., and I.R. are calculated. If the level of inconsistency present in the information stored in the “D” matrix i s satisfactory, the QFD team, then, puts the PV values in the transformation mat rix. The next job of the QFD team is to find out the ranking of the given four r obots based on the seven conflicting TRs. Seven pair-wise comparison matrices we re built up based on the information on each TR. Table 2. Overall Scores of the Four Robots Technical Requirements Weight Importance weight for robots I.I. I.R. Inconsisten cy (%) The PV values R1 31.54 8.64 9.47 9.36 9.05 26.46 5.48 0.529 0.147 0.074 0.267 0.319 0.523 0.48 3 40.53 R2 0.094 0.281 0.520 0.550 0.532 0.089 0.086 23.11 R3 0.314 0.514 0.105 0.054 0.092 0.326 0.355 27.25 R4 0.063 0.059 0.300 0.128 0.057 0.062 0.077 9.11 0.0249 0.0116 0.0842 0.0644 0.

0866 0.0369 0.0748 0.0252 0.0117 0.0851 0.0651 0.0875 0.0373 0.0756 1. Drive system 2.Geometrical dexterity 3. Path measuring system 4. Robot size 5 . Material of robot 6. Weight of robot 7. Initial operating cost Overall score 2.52 1.17 8.51 6.51 8.75 3.73 7.56

312 A. Abraham et al. Table 2 suggests R1 » R3 » R2» R4; i.e., R1 gets precedence over R3, which gets more i mportance over R2 and R4. Thus, the robot R1 is selected as it has the highest o verall score compared with others. The total cost of the robotic system describe d in Bhattacharya et al. (2005) were broken down (refer to Table 3). Table 3. Cost Factor Components and Their Units Cost factor components 1. Acquisition cost of robot 2. Cost of robot gripper mec hanisms 3. Cost of sensors 4. Total cost of layout necessary for installation of robot 5. Cost of feeders 6. Maintenance cost 7. Cost of energy Range of attribu te values US $ 4500 – 7000/unit US $ 2500 – 3000 US $ 900 – 1200 US $ 3500 – 4000 US $ 4 00 – 900/unit US $ 500 – 650/week US $ 6 – 10/Unit of electrical energy The cost factors in Table 3 involve two types of costs, both a fixed and a recur ring type. For four different robots, of which each can perform the very specifi ed job, the attributes of the cost components are tabulated in Table 4. Table 4. Attributes of Cost Factor Component Robots R1 Cost components 1. Acquisition cost of robot 2. Cost of robot gripper mechanisms 3. Cost of sensors 4. Total cost of layout 5. Cost of feeders 6. Main tenance cost 7. Cost of energy Total (OFC) (US$) 6500 2750 1200 3650 900 480 7 1 5487 R2 5000 2500 950 4000 765 900 8 14123 R3 7000 3000 1100 3875 400 730 10 161 15 R4 4500 2900 1000 3500 860 400 6 13166 A mathematical model was proposed by Bhattacharya et al. (2005) to combine cost factor components with the importance weightings found from AHP. The governing e quation of the said model is SIi = [ ( × SFMi ) + ( 1 ) × OFMi ] (7) where,

Neuro-Fuzzy Approximation of MCDM QFD 313 OFM i OFC i 1 n . (8) OFC i 1 1 In the following chapters, we have discussed the implications of Eq. (8) as well as the modified S-curve MF with reference to the targeted MCDM modeling. Theref ore, we refrain to discuss on these basic equations. 5.1 Computation of Level-of-Satisfaction, Degree of Fuzziness We confine our efforts assuming that differences in judgmental values are only 5 %. Therefore, the upper bound and lower bound of SFMi as well as SIi indices are to be computed within a range of 5% of the original value reported by Bhattacha rya et al. (2005). In order to avoid complexity in delineating the technique pro posed herein, we have considered, 5% measurement. One can fuzzify the SFMi value s from the very beginning of the AHP-QFD model by introducing a modified S-curve MF in AHP, and the corresponding fuzzification of SIi indices can also be carri ed out using their holistic approach. By using the equations above for a modifie d S-curve MF a relationship among the level-of-satisfaction of the decision make r, the degree of vaguenes and the SI indices is found. The results are plotted a ccordingly. Figures 5a, b and c show three different plots depicting a relation among the level-of-satisfaction and SI indices for three different vagueness val ues. It should always be noted that higher the fuzziness, , values, the lesser w ill be the degree of vagueness inherent in the decision. Therefore, it is unders tood that the higher level of outcome of the decision variable, SI, for a partic ular level-of-satisfaction point, results in a lesser degree of fuzziness inhere nt in the said decision variable. A relationship between the degree of fuzziness , , and the level-ofsatisfaction has been depicted by Figure 6. This is a clear indication that the decision variables, as defined in Eqs. (6) and (7), allows t he MCDM model to achieve a higher level-of-satisfaction with a lesser degree of fuzziness. Figures 7 and 8 delineate SI indices versus level-of-satisfaction and SI indices versus degree of fuzziness , respectively. Now, let us examine the f uzziness inherent in each candidate-alternative. There is a need to calculate bo th the upper bound and the lower bound solution of SI indices having a different level-of-satisfaction ( ). The

314 A. Abraham et al. following figures have been found using MATLAB® version 7.0. The results have been encouraging, and the corresponding results have been indicated in Figures 9 to 14. Figure 6. Fuzziness vs. for Robot 1 Figure 7. Fuzziness vs. for Robot 2 Figure 8. Fuzziness vs. for Robot 3 Figure 9. Fuzziness vs. for Robot 4 Figure 10. SI vs. for Robot 1 Figure 11. SI vs. for Robot 2

Neuro-Fuzzy Approximation of MCDM QFD 315 Figure 12. SI vs. for Robot 3 Figure 13. SI vs. for Robot 4 Figure 14. SI, , and for Robot 1 Figure 15. SI, , and for Robot 2 Figure 16. SI, , and for Robot 3 Figure 17. SI, , and for Robot 4

316 A. Abraham et al. Thus, the decision for selecting a candidate-alternative as seen from Figures 9 to 13 is tabulated in Table 5. It is noticed from the current investigation that this model eliciting the degree of fuzziness corroborates the MCDM model withou t fuzzification presented in Bhattacharya et al. (2005). 5.2 Experiment Results using the ANFIS Model The experimental system consists of two stages: network training and performance evaluation. The task is to approximate the values of SI for different values of and . In this chapter, we developed fuzzy inference systems for varying values of gamma keeping = 0.001, 0.2, 0.4, 0.6, 0.8, and 1.0. Takagi Sugeno fuzzy infer ence was used with linear consequent parameters. We used four Gaussian MFs for t he two variables and . Sixteen fuzzy if then rules were created during the neura l learning process as depicted in Figures 18, 20, 22, 24, 26 and 28. The learned surfaces showing the input/output are illustrated in Figures 19, 21, 23, 25, 27 and 29. Empirical results are depicted in Table 5. Table 5. Performance of the Fuzzy Inference Systems value 0.001 0.2 0.4 0.6 0.8 1.0 Root Mean Squared Error 0.0004 0.0009 0.0004 0.0 02 0.002 0.004 Figure 18. Developed Takagi Sugeno FIS ( = 0.001)

Neuro-Fuzzy Approximation of MCDM QFD 317 Figure 19. Input/Output surface mapping ( = 0.001) Figure 20. Developed Takagi Sugeno fuzzy inference system ( = 0.2) Figure 21. Input/Output surface mapping ( = 0.2)

318 A. Abraham et al. Figure 22. Developed Takagi Sugeno fuzzy inference system ( = 0.4) Figure 23. Input/Output surface mapping ( = 0.4) Figure 24. Developed Takagi Sugeno fuzzy inference system ( = 0.6)

Neuro-Fuzzy Approximation of MCDM QFD 319 Figure 25. Input/Output surface mapping ( = 0.6) Figure 26. Developed Takagi Sugeno fuzzy inference system ( = 0.8) Figure 27. Input/Output surface mapping ( = 0.8)

320 A. Abraham et al. Figure 28. Developed Takagi Sugeno fuzzy inference system ( = 1.0) Figure 29. Input/Output surface mapping ( = 1.0) 6. DISCUSSION AND CONCLUSION One underlying assumption of the proposed methodology is that the selection is m ade under certainty of the information data. In reality, the information availab le is highly uncertain and sometimes may be under risk also. The fuzzy S-curve M F helps in reducing the level of uncertainty as

Neuro-Fuzzy Approximation of MCDM QFD 321 validated further square errors as are very low, and ciable as well as by introducing the ANFIS model shown in Table 5. The root mean compared with the original values of level-ofsatisfaction ( ) the satisfaction level of the decision makers are, thus, appre within the acceptable level.

REFERENCES Abraham A., 2005, Adaptation of fuzzy inference system using neural learning, fu zzy system engineering: theory and practice. In: Nedjah, N. et al. (eds.), Studi es in Fuzziness and Soft Computing, pp. 53–83, Springer Verlag, Germany. Bhattacha rya, A., Sarkar, B., and Mukherjee, S.K., 2005, Integrating AHP with QFD for rob ot selection under requirement perspective, International Journal of Production Research, 43(17): 3671–3685. Chuang, P.T., 2001, Combining the analytic hierarchy process and quality function deployment for a location decision from a requireme nt perspective, International Journal of Advanced Manufacturing Technology, 18: 842–849. Cohen, L., 1995, Quality Function Deployment – How to make QFD Work for You , Addison – Wesley, New York. France chini, F., and Rossetto, S., 1995, QFD: the p roblem of comparing technical/engineering design requirements, Research Engineer ing Design, 7: 270–278. Govers, C.P.M., 2001, QFD not just a tool but a way of qua lity management, International Journal of Production Economics, 69(2): 151–159. Ha user, J.R., and Clausing, D., 1988, The house of quality, Harvard Business Revie w, May – June: 63–73. Jang, J.S.R., 1991, ANFIS: adaptive network based fuzzy infere nce systems, IEEE Transactions Systems, Man & Cybernetics, 23: 665 685. Saaty, T .L., 1994, How to make a decision: the analytic hierarchy process, Interfaces, 2 4(6): 19–43. Saaty, T.L., 1990, How to make a decision: the analytic hierarchy pro cess, European Journal of Operational Research, 48(1): 9–26. Saaty, T.L., 1988, Th e Analytic Hierarchy Process, Pergamon, New York. Saaty, T.L., and Vargas, L.G., 1987, Uncertainty and rank order in the analytic hierarchy process, European Jo urnal of Operational Research, 32: 107–117. Saaty, T.L., 1980, The Analytical Hier archy Process, McGraw-Hill, New Work. Sugeno, M., 1985, Industrial Applications of Fuzzy Control, Elsevier Science Pub Co., New York. Sullivan, L. P., 1986, Qua lity function deployment, Quality Progress, 19(6): 39–50. Wasserman, G.S., 1993, O n how to prioritize design requirements during the QFD planning process, IEEE Tr ansactions, 25(3): 59–65.

FUZZY MULTIPLE OBJECTIVE LINEAR PROGRAMMING Cengiz Kahraman and hsan Kaya Department of Industrial Engineering, Istanbul Technical University, Maçka, Istanb ul, Turkey Abstract: In this chapter, first a literature review on the fuzzy multi-objective linear p rogramming (FMOLP) and then its mathematical modeling with an application is giv en. FMOLP is one of the multi-objective modeling techniques most frequently used in the literature. The possible values of the parameters in FMOLP are imprecise ly or ambiguously known to the experts. Therefore, it would be more appropriate for these parameters to be represented as fuzzy numerical data that can be repre sented by fuzzy numbers. Multiple objectives, linear programming, interactive, a pproximation algorithm Key words: 1. INTRODUCTION Multiple objective problems are concerned with the optimization of multiple, con flicting, and noncommensurable objective functions subject to constraints repres enting the availability of multiple objective problems that are concerned with t he optimization of multiple, conflicting, and noncommensurable objective functio ns subject to constraints representing the availability of limited resources and requirements. Multiple objective linear programming (MOLP) is one of the popula r methods to deal with complex and ill-structured decision problems. When formul ating an MOLP problem, various factors of the real world should be reflected in the description of the objective functions and the constraints. Naturally, these objective functions and constraints involve many parameters in which possible v alues may be assigned by the experts. C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 325

326 C. Kahraman and . Kaya Normally, such parameters are set at some values in an experimental or subjectiv e manner through the experts’ understanding of the nature for the parameters (Saka wa, 1993). The MOLP problem is specified by linear functions that are to be maxi mized subject to a set of linear constraints. The standard form of MOLP can be w ritten as follows: Maximize f x Cx (1) x R Ax n subject to x X b, x 0 where C~is an k n objective function matrix, A is an m n constraint matrix, b is an m-vector of the right-hand side, and x is an n-vector of decision variables. With this observation, it is natural to recognize that the possible values of t hese parameters are often imprecisely or ambiguously known to the experts. In th is case, it may be more appropriate to interpret the experts’ understanding of the parameters as fuzzy numerical data that can be represented by fuzzy numbers. Th e fuzzy multiple objective linear programming (FMOLP) problems involving fuzzy p arameters would be viewed as a more realistic version than the conventional one (Sakawa, 1993). Various kinds of FMOLP models have been proposed to deal with di fferent decision-making situations that involve fuzzy values in objective functi on parameters, constraints parameters, or goals. Tanaka and Asai (1984) formulat ed FMOLP with triangular fuzzy numbers, and the nonlinear programming problem ob tained was solved by using a max–min operator. Luhandjula (1987) proposed the conc ept of -possible feasibility and -possible efficiency and resolved imprecise obj ectives and constraints with fuzzy numbers by solving an auxiliary crisp MODM pr oblem derived by using the extension principle and -and -level cuts. Korhonen et al. (1989) propose a general approach to semistructured decision making, which makes it possible to consider multiple objectives (flexible gloals), “hard” constrai nts (inflexible: goals), and “soft” constraints (fuzzy goals) within the same framew ork. Rommelfanger et al. (1989) present a new method calleld “alpha-level related pair formation” for solving linear programming problems with fuzzy parameters in t he objective function. Lai and Hwang (1992) resolved imprecise objectives with t riangle fuzzy numbers with maximizing the most possible value, minimizing the ri sk of obtaining lower profit, and maximizing the possibilities of obtaining high er profit, and they used a fuzzy ranking

Fuzzy Multiple Objective Linear Programming 327 concept to resolve imprecise constraints. Slowinski and Tenghem (1993) compare t he methods fuzzy linear programming (FLIP) and strategy for nuclear generation o f electricity (STRANGE) developed by themselves, respectively. Zimmermann (1993) surveys major models and theories in mathematical programming and offers some i ndication on the expected future developments. Turtle et al. (1994) show how fuz zy logic can be employed using straightforward LP tools. Julien (1994) investiga tes the application of fuzzy set and possibility theories for the representation of imprecise information in water quality management problems. Fuller and Fedri zzi (1994) explore stability analysis in possibilistic programming by extending previous research results to possibilistic linear programs with multiple objecti ve functions. They use multi-objective possibilistic linear programs with contin uous fuzzy number coefficients. Nakahara and Gen (1994) propose a quantitative f ormulation of LP problems with fuzzy number coefficients, by using the ranking c riteria proposed by themselves, and show an algorithm for solving the formulated problems in some cases. The optimization of an objective function with fuzzy nu mber coefficients is formulated as the user-oriented extension of the optimizati on of an objective function with real coefficients by the proposed ranking crite ria. Carlsson and Fuller (1995) introduce measures of interdependence between th e objectives in order to provide for a better understanding of the decision prob lems and to find effective and more correct solutions into multiple criteria dec ision-making problems. Sakawa et al. (1995) show that large-scale fuzzy LP probl ems can be reduced to a number of independent linear sub-problems (and the overa ll satisfying solution for the decision maker is directly obtained just solving the sub-problems. Herrera and Verdegay (1995) study some models for dealing with fuzzy integer LP problems that have a certain lack of precision of a vague natu re in their formulation and present methods to solve them with either fuzzy cons traints or fuzzy numbers in the objective function or fuzzy numbers defining the set of constraints. Kahraman et al. (1996) propose a fuzzy multi-objective line ar programming that considers intangible benefits in AMTs and expands the constr aints by adding tolerances. The transition from vagueness to quantification is p erformed by applying the fuzzy set theory. The approach also considers the vague ness in the objective functions by using the membership functions. The main adva ntage of the fuzzy LP, compared with the unfuzzy problem formulation, is the fac t that the decision maker is not forced into a precise formulation for mathemati cal reasons. Downing and Ringuest (1998) implement four multi-objective linear p rogramming algorithms on microcomputer software packages and

328 C. Kahraman and . Kaya conduct a large field experiment using the implemented algorithms. Two new algor ithms that incorporate formal models of decision maker behavior are tested along with two established algorithms that include no formal models of decision-maker behavior. Borges and Antunes (2002) study the effects of uncertainty on multipl e objective linear programming models using the concepts of fuzzy set theory. Th e proposed interactive decision support system is based on the interactive explo ration of the weight space. The comparative analysis of indifference regions on the various weight spaces (which vary according to intervals of values of the sa tisfaction degree of objective functions and constraints) enables study of the s tability and evolution of the basis that correspond to the calculated efficient solutions with changes of some model parameters. Wang and Liang (2004) develop a n FMOLP model for solving the multi-product aggregate production planning (APP) decision problem in a fuzzy environment. The proposed model attempts to minimize total production costs, carrying and backordering costs, and rates of changes i n labor levels considering inventory level, labor levels, capacity, warehouse sp ace, and the time value of money. Jana and Chattopadhyay (2004) design a model o f energy utilization by developing a decision support frame for an optimized sol ution to the problem, taking into consideration four sources and six devices sui table for the study area, namely Narayangarh Block of Midnapore District in Indi a. Since the data available from rural and unorganized sectors are often illdefi ned and subjective in nature, many coefficients are fuzzy numbers, and hence sev eral constraints appear to be fuzzy expressions. In this study, the energy alloc ation model is initiated with three separate objectives for optimization, namely minimizing the total cost, minimizing the use of nonlocal sources of energy, an d maximizing the overall efficiency of the system. Since each of the above objec tive-based solutions has relevance to the needs of the society and economy, it i s necessary to build a model that makes a compromise among the three individual solutions. El-Ela et al. (2005) present a proposed procedure that depends on the multi-objective fuzzy linear programming (MFLP) technique to obtain the optimal preventive control actions, for power generation and transmission line flows, t o overcome any emergency conditions. The proposed multiobjective functions are m inimizing the generation cost function, maximizing the generation reserve at cer tain generator, maximizing the generation reserve for all generation system, and maximizing the preventive action for one or more critical transmission line.

Fuzzy Multiple Objective Linear Programming 329 Jana and Roy (2005) present the solution procedure for a multiobjective fuzzy li near programming problem (MOFLPP) with mixed constraints and its application in solid transportation problem. There are two parts in this paper. In the first pa rt, a multi-objective linear programming problem with fuzzy coefficients occurri ng in constraints and objective functions and fuzzy constraint goals is consider ed. Fuzzy constraint goals and coefficients of objective and constraint function s are characterized by triangular fuzzy numbers (TFNs). Using Bellman and Zadeh’s (1970) multi-criteria fuzzy decision-making process, the very problem is convert ed to a crisp non linear programming problem. Then it is solved using a fuzzy de cisive set method. In the other part, a linear multi-objective solid transportat ion problem with mixed constraint as well as an additional restriction in a fuzz y environment is considered. In this transportation problem, the cost coefficien ts of objective functions and the additional restriction function as well as the supply, demand, and conveyance capacity are expressed as TFNs. This MOFLPP is s olved by the fuzzy decisive set method as in the first part. Wu et al. (2006) de velop a new approximate algorithm for solving FMOLP problems involving fuzzy par ameters in any form of membership functions in both objective functions and cons traints. Liang (2006) develops an interactive fuzzy multi-objective linear progr amming (i-FMOLP) method for solving the fuzzy multi-objective transportation pro blems with a piece-wise linear membership function. The interactive FMOLP method includes the following steps (Liang, 2006): Step 1. Formulate the original fuzz y MOLP model for the considered problem. Step 2. Given the minimum acceptable me mbership level, , and then convert the fuzzy inequality constraints with fuzzy a vailable resources (the right-hand side) into crisp ones using the weighted aver age method. Step 3. Specify the degree of membership for several values of each objective function. Step 4. Draw the piece-wise linear membership functions for each objective function. Step 5. Formulate the piece-wise linear equations for e ach membership function. Step 6. Introduce an auxiliary variable, thus enabling the original fuzzy multi-objective problem to be aggregated into an equivalent o rdinary LP form using the minimum operator.

330 C. Kahraman and . Kaya Step 7. Solve the ordinary LP problem, and execute the interactive decision proc ess. If the decision maker is dissatisfied with the initial solutions, the model must be adjusted until a set of satisfactory solutions is derived. Li et al. (2 006) improve the fuzzy compromise approach of Guu and Wu (1999) by automatically computing proper membership thresholds instead of choosing them. In practice, c hoosing membership thresholds arbitrarily may result in an infeasible optimizati on problem. Although they can adjust minimum satisfaction degree to get a fuzzy efficient solution, it sometimes makes the process of interaction more complicat ed. In order to overcome this drawback, a theoretically and practically more eff icient twophase max–min fuzzy compromise approach is proposed. 2. FUZZY MULTI-OBJECTIVE LINEAR PROGRAMMING When all coefficients of the objective functions and the constraints are fuzzy n umber parameters represented in any form of membership functions, such FMOLP pro blems can be formulated as follows: ~ Maximize f x ~ Cx x n ~ R Ax (2) ~ b,x 0 subject to x X ~ ~ where C is an k n matrix, each element of which cij is a fuzzy ~ ~ij x ; A is an number c~ij x , represented by membership function c ~ m n matrix, each eleme nt of which aij is a fuzzy number represented by ~ ~ membership function aij x ; b is an m-vector, each element of which ~ ~ bi is a fuzzy number represented by membership function bi x ; and x is n an n-vector of decision variables, x R . Associated with the FMOLP problems (Eq. 2), the following MOLP problems can be w ritten as (Wu et al., 2006):

Fuzzy Multiple Objective Linear Programming CLx Maximize subject to x X x R n AL x b L , AR x bR , x 0, 0,1 CRx , 0,1 331 (3) where cL 11 cL 21 cL k1 cR 11 cR 21 cR k1 L a 11 L a 21 a L m1 CL cL 12 cL 22 cL k2 cR 12 cR 22 cR k2 L a 12 L a 22 a L m2 cL 1n cL 2n cL kn cR 1n cR 2n cR kn L a 1n L a 2n a L mn CR A L A R R a 11 R a 21 a R m1 R a 12 R a 22 a R m2 R a 1n R a 2n a R mn

332 C. Kahraman and . Kaya bL bR b1L , b2L , R R b1 , b2 , L , bm T R , bm T Wu et al. (2006) propose an approximation algorithm as follows for solving MOLP problem, the solution of which is equally the solution of FMOLP problem. Maximize (4) subject to ci fi L j x fi L max j L min j L min fi j , ci fi R j x fi R max j R min j R min fi j L as j x bsL j , R as j x bsR j where

fi Lmax j fi Rjmax ciL j xiL*j , i 1, 2,..., k ; j ciR j xiR*j , i 1, 2,..., k ; j f i L min j s 1 ,...,k t 1 ,...,l s i ,t j 0,1,..., l , 0,1,..., l , L R min ciL j x s *t ,ciL j x s *t f i R jmin s 1 ,...,k t 0 ,1 ,...,l s i ,t j R L R min ci j x s *t ,ciR j x s *t and x* is said to be an optimal solution. 2.1 A Numerical Example Assume that you have two fuzzy linear objective functions and four fuzzy linear constraints as follows.

Fuzzy Multiple Objective Linear Programming 333 Max f x max f1 x f2 x c11 x1 c12 x2 c21 x1 c22 x2 subject to a11 x1 a12 x2 b1 , a21 x1 a22 x2 b2 0, x x2 c11 1, 7 5 25 24, 5 x 9 19, 9 x 10 x 7 100 x 2 0, 10 x 0, x 2 x 7 x 2 5, 2 c12 1, 7 x 12 196 x 2 0, 14 x 52, 12 x 14 0, x 14 x2 c21 1, 18 196 188, 14 x 22 92, 22 x 24 x 18 576 x 2 0, 24 x

334 C. Kahraman and . Kaya 0, x x2 c22 1, 35 30 900 325, 30 x 40 425 40 x 45 x 35 2025 x2 0, 45 0, x 2 x, 0 a11 1, 0.5 x 0 x x 0.5 2 x 5 25 5 x 15, 2 0, 5 0, x 5 x, 0 1, 0.2 x 0 x 0.2 x 3 x 1 a12 9 3 x 6, 1 0, 3 0, x 4 x, 0 x 0 x x 0.25 5 x 8 a21 1, 0.25 64 8 x 24, 5 0, 8 x

Fuzzy Multiple Objective Linear Programming 335 0, x 5 x 6 2 x 10 2 , 5 a22 1, 6 x 7 64 8 x 8, 7 0, 8 x x 8 0, x 12 x 12 3, 12 b1 1, 15 x 18 x 21 x 15 42 2 x 6, 18 0, 21 0, x x 56 2 x 112 24, 56 b2 1, 68 x 74 x 68 258 3 x 36 , 74 0, 86 x x 86 This fuzzy MOLP problem can be solved by the normal Simplex algorithm. 3. CONCLUSIONS Multiple objective problems are concerned with the optimization of multiple, con flicting, and noncommensurable objective functions subject to constraints repres enting the availability of limited resources and requirements. In this chapter a n example of FMOLP is provided in which all coefficients are the objective funct ions and the constraints are fuzzy number parameters represented in any form of membership functions. The fuzzy multi-objective LP provides great flexibility to make the estimates of the problem parameters. The main advantage of the fuzzy L P, compared with the unfuzzy problem formulation, is the fact that the decision maker is not forced into a precise formulation because of mathematical reasons.

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QUASI-CONCAVE AND NONCONCAVE FMODM PROBLEMS Chian-Son Yu1 and Han-Lin Li 2 1 Graduate Institute of Business Administration, Department of Information Managem ent, Shih Chien University, Taipei, Taiwan 2School of Management, Institute of I nformation Management, National Chiao Tung University, Hsinchi, Taiwan Abstract: A membership function may be concave-shaped or convex-shaped. In this chapter, f irst, concave and convex membership values are analyzed and, in practice, common ly used approaches for solving an fuzzy multi-objective decision-making (FMODM) problem are briefly reviewed. Then, some proposition and remarks are presented t o solve a quasi-concave FMODM problem. The proposed method can directly solve a quasi-concave FMODM problem by using standard LP techniques. Quasi-concave, nonc oncave, LP Key words: 1. INTRODUCTION Decision making (DM) is part of people lives, and the history of DM even dates b ack to before the dawn of history, but the scientific research of a systematic p rocedure of describing the human DM process appeared in the early 1960s (Simon, 1960). Although the DM theory proposed by Simon has received a lot of attention, the DM process described by Simon lies on single objective only. In many real d ecision situations, more than one objective has to be considered and different k inds of uncertainty must be handled (Abdelaziz et al., 2004), particularly in mu lti-dimension, multi-criterion, and nondominated perspective DM problems. Theref ore, multi-objective decisionmaking (MODM) programming have long drawn a wide sp ectrum of attention from both academicians and practitioners. Typically, MODM st arts with determining a set of objectives and ends with finding the best accepta ble C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 339

340 C.-S. Yu and H.-L. Li solution based on the decision-maker preferences. Examples of MODM include purch asing a car, recruiting a new manager, choosing the best portfolio, appointing a spokesperson, or underwriting an insurance contract. Because of human beings’ inh erent subjectivity, imprecision, and vagueness in expressing judgments, in pract ice, decision makers may frequently express their evaluations in a form of uncer tainty rather than preciseness. Since fuzzy theory is very helpful in dealing wi th fuzziness of human judgment quantitatively, using fuzzy theory to treat MODM has been discussed since the 1970s (Yu, 2001). However, fuzzy multi-objective de cision making (FMODM) problems have been noticed worldwide since the work publis hed by Zimmermann (1981) where Zimmermann introduced conventional linear program ming and multi-objective linear programming into fuzzy set theory. Since then, v arious methods using linear programming (LP) have been developed to solve FMODM problems. An FMODM problem is usually formulated to maximize and/or minimize sev eral objectives simultaneously subject to a constraint set. Accordingly, a gener al FMODM problem, in which the aggregated goal is the minimum operator of indivi dual goals, is formulated as follows: FMODM Problem: Maximize (1.1) subject to i ( z i ) , i = 1, 2, ..., n F (a feasible set), i ( z i ) = z i ( X ) g i , z i ( X ) where i ( zi ) is the membership function of the ith objective function , g i denotes the fuzzy goal of the ith objective function, zi ( X ) is the ith objective function, and X is a vector of decision variables. Many studies (Biswa l, 1997; Hannan, 1981a, 1981b; Mjelde, 1983; Nakamura, 1984; Narasimhan, 1980; R omero, 1994; Yang et al., 1991) indicate that most real-world applications in en gineering, physical, business, social, and management fields are not pure linear , triangular, concave, or convex FMODM problems but rather quasi-concave or more general nonconcave FMODM problems. Due to one of the most promising techniques of linearizing non concave functions is piece-wise linear programming. Hence, FM ODM problems with piece-wise linear membership functions have been studied by Na rasimhan (1980), Hannan (1981), Nakamura (1984), Inuiguchi et al. (1990), and Ya ng et al. (1991). A membership function i ( zi ) may be concave-shaped or convex shaped, as shown in Figure 1(a) and (b), respectively. The marginal possibility with respect to a concave membership function is decreasing,

Quasi-Concave and Nonconcave FMODM Problem 341 whereas the marginal possibility with respect to a convex membership function is increasing. If the marginal possibility increases first, then decreases, or dec reases first then increases, then the membership function becomes a convex–concave or concave convex mixed shape as shown in Figure 1(c). In practice, membership functions are not concave or convex but the mixed shapes composed by concave and convex curves or even more general non concave curves as shown in Figure 1(d). 2. REVIEW OF FMODM PROGRAMMING Commonly used approaches for solving a FMODM problem in Eq. 1 are briefly review ed in this section. In 1980, Narasimham proposed a LP approach to solving a FMOD M problem with triangular membership functions, as expressed below. FMODM Model 1 (Narasimham Method): Maximize subject to i ( zi ) , i = 1, 2, ... , n if g i bi gi di di bi gi di bi bi di gi di 0 1 i gi bi if b i if b i (1) ( zi ) 1 0 otherwise zi ( X ) F (a feasible set), where i ( z i ) is the membership function of the i th objective function, g i denotes the ith fuzzy goal, z i ( X ) is the ith obje ctive function, X is a vector of decision variables, d i is a chosen positive co nstant for the maximum allowable deviation from the aspiration level of the ith goal, and bi is the most desired value of the ith objective function. Two primar y drawbacks in Narasimham’s method are listed below: 1. An FMODM problem has to be divided into 2n sub-problems where n is the number of goals. 2. All membership functions are restricted in triangular or trapezoidal shapes. Extending triangul ar or trapezoidal to general concave-shaped membership functions, Hannan (1981) presented a method for converting a FMODM problem in Eq. 1 into the following mo del:

342 i(zi) C.-S. Yu and H.-L. Li zi(X) (a) A concave membership function i(zi) zi(X) (b) A convex membership function i(zi) zi(X) (c) A concave-convex mixed membership function i(zi) a1 a2 a3 a4 zi(X) (d) A more general non-concave membership function Figure 1. Membership Functions FMODM Model 2 (Hannan Method): Maximize subject to (2) i ( zi ) , i = 1, 2, ..., n zi d ij d ij g ij ,

Quasi-Concave and Nonconcave FMODM Problem 343 zi – F , d ij + 0, d ij 0, j = 1, 2, ..., N i where i ij i ( zi Ni ), a concave typed function, is expressed as ij ( zi ) j 1 i zi g ij i i z ri g ij d ij in which , , a n d ri are parameters, g ij are the change points of segments, + d ij and d ij are deviation variables. The serious limitation in Hannan’s method is that all i ( zi ) should be concave f unctions. For tackling a quasi-concave FMOP problem, Inuiguchi et al. (1990) dev eloped a approach of transforming all quasiconcave functions into concave functi ons. Consider the following example slightly modified from Inuiguchi et al. (199 0). Example 1: Maximize subject to 1 ( z1 ) , 2 zi d ij

( z2 ) , z1 = x1 + 2x2, z2 = 2x1 + x2, x1 + 3x2 21, x1 + 3x2 27, 4x1 + 3x2 45, 3x1 + x2 3 0, x1 , x2 0 0, 0.04 z1 , 1 ( z1 z1 3 0.2, 2.2, 0.5, 2 z1 12 17 z1 z2 7 0 .6, 1 .7 , 0 .8, 0 .5, 17 z2 21 27 30 z2 3 z1 z1 12 z1 z1 27 7 z2 z2 21 z2 z2 z2 32 27 30 32 17 21 2 12 17 27 )= 0.08 z1 1, 0.1 z1 0.05 z1 0, 0, 0 .0 6 z 2 , 0 .1 z 2 1, 0 .0 3 3 z 2 0 .1 z 2 0 .2 5 z 2 0, 2 ( z2 )=

344 C.-S. Yu and H.-L. Li 1 where ( z1 ) and 2 ( z2 ) are specified in Figure 2(a). 1(z1) 2(z2) 1.0 0.033 0.8 0.08 0.6 0.5 0.4 0.2 3 0.04 2 (a) 7 1(z1) 0.1 0.1 0.1 0.06 0.05 0.25 12 and 17 2(z2) 21 27 in Example 1 30 32 1(z1) 2(z2) 1.0 0.8 0.769 0.5 0.385 3 7 3/3 12 1(z1) 21 and 2(z2) 27 91/3 32 (b) Two converted in Inuiguchi et al. Model Figure 2. Membership values

Notably both 1 ( z1 ) and 2 ( z2 ) are quasi-concave functions, as depicted in F igure 2(a). Inuiguchi et al. first convert 1 ( z1 ) and 2 ( z2 ) into two concav e functions 1 ( z1 ) and 2 ( z2 ) , respectively, as shown in Figure 2(b). Examp le 1 then can be solved by the following LP model: FMODM Model 3 (Inuiguchi et a l. Method for Example 1):

Quasi-Concave and Nonconcave FMODM Problem 345 Maximize subject to z1 2 ( z2 ) 1 ( z1 ) , x1 2 x2 , z2 2 x1 x2 x1 3 x2 21 , x1 3x2 27 0 3 3 z1 z1 12 z1 27 27 12 4 x1 3x2 0, 1 ( z1 45 , 3x1 1 z1 13 9 , 5 3 , 13 x2 3 z1 65 30 , x1 , x 2 z1 29 65 , )= m in 1, 1 z1 15 0, 12 z1 0, 2 ( z2 z2 3 z2 26 1 z2 5 21 3 , z2 26 52 32 , 3 1 z2 15 11 , 52 8 , 3 1 z2 45 53 , 45 7 z2 21 z2 7 z2 21 z2 32. 32 17 ) = m in 1, m in 0, Although Inuiguchi et al.’s idea is very useful in formulating quasiconcave functi ons into concave functions, there are three shortcomings in Inuiguchi et al. met hod as described below: If the number of break points is large, then it causes a tedious computational burden to convert these membership functions into concave functions. That transforming procedure is complicated and cannot effectively de al with an FMODM problem with more general nonconcave functions. That method sti ll requires zero-one variables to treat converted concave functions [i.e., 1 ( z 1 ) and 2 ( z2 ) ]. Take Example 1, for instance: Five break points are required to do transforming computing. Suppose there are n objective functions and each of these functions having mi break points, then the number of transforming compu ting is n mi i 1

The situation would become more complicated for treating more general nonconcave FMODM problems. Consequently, Yang et al. (1991)

346 C.-S. Yu and H.-L. Li presented another method for treating a quasi-concave FMODM problem. Take Exampl e 1, for instance. Yang et al.’s method could formulate Example 1 as following a z ero-one programming model [as depicted in Figure 3(a) and 3(b)]: FMOP Model 4 (Y ang et al.’s method for Example 1) Maximize subject to 12 z1 1 d2 1 M a4 1 z1 d1 M 2 M (1 , 1 ) M 2 1 1, a3 d3 1 a6 z1 z2 d5 1 a9 M 1 M 2 1 1 1 27 z1 d4 21 z2 d6 32 z2 d9 M (1 M M , 2 ) M 1 M (1 z2 d8 M 3 ) 3

a10 z2 d7 M 3, 3 3, z1 x1 2 x2 21 45 0 z2 2 x1 x2 , 27 , 30 , x1 3 x2 4 x1 3x2 x1 , x 2 x1 3x2 3x1 x2 where 1 , 2 , and 3 are zero one variables; M is a big number; and a1, a2, …, a10 are approximated end-point values as depicted in Figure 3(a) and 3(b). A major d isadvantage in Yang et al.’s method (1991) is that it involves too many zero-one v ariables for treating quasi-concave FMODM problems. The number of zero-one varia bles equals the number of intersections between convex and concave functions. In addition, many end-point approximations are required before formulating a quasi -concave FMODM program. Take Example 1, for instance, i ( zi ) contains two conv ex concave intersections and 2 ( z 2 ) contains one convex concave intersection. Therefore, three zero one variables (i.e., 1 , 2 , or 3 ) are added in the solu tion process. In addition, ten times end-point approximations (i.e., a1, a2, …, a1 0) are required in formulating FMODM model 2. A detailed discussion is given in Li and Yu (1999). Considering i ( zi ) in Problem (1.1) could be concave, convex , or concave-convex mixed type functions, Nakamura developed a method to

Quasi-Concave and Nonconcave FMODM Problem 1 ( z1 ) 347 1.0 0.5 0.2 z1 ( X) 3 2 12 17 27 a1 a2 d1 d2 (a) a3 a4 d3 d4 (z1) in Yang et al. Model 1 2 ( z2 ) 1.0 0.6 0.5 0 7 a5 d5 17 d6 a6a7 27 a8 d8 30 32 d7 a9 a10 d9 (b) 2 ( z2 ) in Yang et al. Model Figure 3. Membership Values expres a general piece wise membership function (1984). He reformulates Problem (1.1) as follow: FMODM Model 5 (Nakamura Method):

348 C.-S. Yu and H.-L. Li Maximize subject to where ~ D ( zi ~ D ( zi im ) for i = 1, 2, ..., n im j ( zi im j ( zi ) [{( j 1 )) j 1 )} { j 1 j ( zi )} 1 ] 0 in which j = 1, 2, ..., im are the change points over the convex part of each i ( zi ) , j = 1, 2, ..., i m are the number of linear functions for separating concave or co nvex parts over each i ( zi ) , j = 1, 2, ..., im are the change points over con cave part of each i ( zi ) , and i ( zi ) are linear functions representing part of i ( zi ) . Nakamura’s method encounters two major difficulties: Expression of piece-wise membership functions is intricate; it requires repetitive use of an L P computation for solving an FMOP problem. n That method divides an FMOP problem into n i 1 2ki sub-problems and requires 2 ki constraints, where k i is the number of segments for each i 1 i ( zi ) . Take Example 1, for instance, Nakamura expresses the membership functions , depicted in Figure 4(a) and 4(b), as follows: 1 ( z1 ) [{ 1 ( z1 ) 2 ( z1 )} { 1 ( z1 )} { 7 ( z2

3 ( z1 ) ) 4 ( z1 )} 1 ] 0 ) 1] 0 2 ( z2 ) [{ 5 ( z2 ) 6 ( z2 )} 2 ( z2 ) ) 8 ( z2 9 ( z2 stands for maximum, stands for minimum, { 1 ( z1 ) 2 ( z1 )} , { 3 ( z1 ) 6 ( z 2 )} are the sets of the convex 4 ( z1 )} , and { 5 ( z 2 ) parts. where Nakamur a’s method then divides Example 1 into eight sub-problems. Some of these sub-probl ems are expressed as follows: FMODM Model 6 (Nakamura Method for Example 1): Sub -problem 1: Maximize subject to 5 ( z2 ) ( zi ) ( z2 ) 2 1 1 ( z1 ) 7 ( z2 ) 3 ( z1 ) 8 ( z2 ) 9 ( z2 )

Quasi-Concave and Nonconcave FMODM Problem 349 Sub-problem 2: Maximize subject to 5 ( z2 ) 1 ( z1 ) ( zi ) 2 ( z2 ) 2 ( z1 ) 7 ( z2 ) 1 3 ( z1 ) 8 ( z2 ) 9 ( z2 ) 1.0 2 1 (Z1 ) 3 (Z1 ) ( Z1 ) 0.4 0. 2 1 4 (Z1 ) 27 z1 ( X) (Z1 ) 2 12 17 3 (a) 2 ( z2 ) 1 ( z1 ) in Nakamura’s Model 1.0 (Z2) (Z2) 7

0.8 0.6 0.5 6 (Z2) 8 2 (Z2) 9 (Z2) 5 (Z2) z2 ( X) 12 17 21 27 30 32 0 7 (b) 2 ( z2 ) in Nakamura’s Model Figure 4. Membership functions

350 C.-S. Yu and H.-L. Li Sub-problem 6: Maximize subject to 6 ( z2 ) ( zi ) 2 ( z2 ) 2 1 ( z1 ) 7 ( z2 ) 3 ( z1 ) 8 ( z2 ) 9 ( z2 ) : After using LP computation repeatedly, Nakamura’s method finds the optimal solutio n in Sub-problem 6. To solving Example 1, Nakamura’s method involves eight sub-pro blems and finds the optimal solution in Subproblem 6 after using the LP computat ion repeatedly. As a result, Nakamura‘s method encounters two major difficulties: Expression of piece-wise membership functions is intricate; it requires repetiti ve use of the LP computation for solving an FMOP problem. n Nakamura’s method has to divide an FMOP problem into 2 k i subproblems and requires 2 ki constraints, where k i is the number of segments for each i ( zi ) . i 1 n i 1 3. PROPOSED METHOD Building on the above discussion, this section first presents a convenient way t o interpret a piece-wise linear membership function. The proposed expression is simpler than Nakamura’s method (1984). An FMODM problem in Equation (1) with piece -wise quasi-concave functions is termed a quasi-concave FMODM problem. Some prop ositions and remarks, presented by Yu and Li (2000), for solving a quasi-concave FMODM problem are described as follows. PROPOSITION 1. Let i ( zi ) be a piecewise linear membership function of zi ( X ) , as depicted in Figure 5(a), where a k , k = 1, 2, ..., m are the break points of i ( z i ) , sk , k = 1, 2, ..., m -1 are the slopes of line segments between a k and ak 1 , and sk i ( zi i (ak 1 ) i (ak ) ak 1 ak ) can then be expressed as:

Quasi-Concave and Nonconcave FMODM Problem m 1 i 351 ( zi ) = i (a1 ) +s1 ( z i ( X ) a1 ) + k 2 sk sk 2 1 ( z i ( X ) ak zi ( X )

a k ) (3) where o is the absolute value of 0. i(zi) s3 s4 s5 s6 s2 s1 0 a1 a2 a3 a4 a5 …………. sm am+1 1 zi am Figure 5. Membership functions This proposition can be examined as follows: (Proof) (i) If zi ( X ) i a2, then ( zi ) = i (a1 ) + i (a2 ) i (a1 ) ( zi ( X ) a1 ) = a1 s1 ( zi ( X ) a1 ) a2 a1 (ii) If zi ( X ) i a3, then (a1 ) + s1 (a2 a1 ) s2 ( zi ( X ) a2 ) s2 s1 ( zi ( X ) a2 zi ( X ) a2 ) = i

(a1 ) + s1 ( zi ( X ) a1 ) 2 ( zi ) = i m 1 (iii) If zi ( X ) i ( zi ak , then k k ( zi ( X ) ak 0 and ) becomes k 1 a1 ) i ( a1 ) + s1 ( zi ( X ) k 2 sk 1 ( zi ( X ) ak zi ( X ) ak ) 2 zi ( X ) ak ) sk

352 C.-S. Yu and H.-L. Li 1 ( z1 1 ( z1 Take instance, ) and ) and 3 2 ( z2 2 ( z2 ) in Example 1 [as depicted in Figure 2(a)] for ) can be represented by Proposit ion 1 as 0.04 z1 2 z1 2 0.04 z1 1 ( z1 ) 0.08 2 z1

(4) 17 z1 17 0.1 2 0.08 12 z1 12 0.05 0.1 0.06 z 2 7 0.1 0.06 17 z2 17 2 z2 0.033 2 0.1 0.1 2 z 2 21 z 2 21 z2 2 z1 2

0.033 (5) z2 27 0.25 0.1 30 z2 30 An advantage of expressing a quasi-concave membership function by Eq. (3) is the convenience of knowing the intervals of convexity and concavity for i ( zi ) , as described below: Remark 1 (Define a convex-type break point). For a i ( zi ) expressed by Eq. (3), if s k 1 s k , then i ( z i ) is a convex function for ak 1 z i ( X ) ak 1 and ak is called a convex-type break point of zi. Take Eq. (4) for instance, it is convenient to check that 1 ( z1 ) is concave when 2 z1 ( X ) 17 and that 1 ( z1 ) is convex when 3 z1 ( X ) 12 and 12 z1 ( X ) 27. Therefore , the point z1 ( X ) 2 and z1 ( X ) 17 are convextype break points of zi. Simila rly for Equation (6), 2 ( z 2 ) is convex for 7 z 2 ( X ) 21 and concave for 17 z 2 ( X ) 32. z 2 ( X ) 17 is a convextype break point of z2. Remark 2 (Define a concave-type break point). For a i ( zi ) expressed by Eq. (3) if s k 1 s k the n i ( zi ) is a concave function for ak 1 z i ( X ) ak 1 and ak is called a conc ave-type break point of zI. Remark 3 (Define a mapping point). For 1 ( z1 ) and 2 ( z 2 ) shown in Figure 6(b) and 6(c), respectively, we can find a convex-type break point bj in z2 by using Remark 1. Then a corresponding point of bj can be found in z1 which has the same value of membership functions as bj. Such a poin t is called a mapping point of bj, denoted as b j , which is mapped from z2 to z 1 and is calculated by b j = 1 1 ( 2 ( b j )) . z2 27 z2 2

Quasi-Concave and Nonconcave FMODM Problem 1 ( z1 ) 2 ( z2 ) 353 z1(X) ai-1 bj = 1 z2(X) bj 1 ai 1 ai+1 2 bj bj+1 ( (bj )) 1 ( z1 ) (a) a concave function (b) a convex function 2 ( z2 ) Figure 6. Concave and convex functions Remark 4 (Specify a converted concave function). Now let us consider two piece-w ise linear functions 1 and 2 specified in Figure 6(a). 1( f ( X )) 1 ( a1 ) s1 ( f ( X ) a1 ) s2 2 s1 ( f ( X ) a 2 (6) (7) 2( f ( X )) 2 ( b1 ) t 1 ( f ( X ) b1 ) t2 2 f ( X ) a 2 )

t1 ( f ( X ) b2 f ( X ) b2 )

where s1 > s2 > 0, t2 > t1 >0, a1 = b1, and a3 = b3. Then two converted concave functions 1 and 2 , shown in Figure 6(b), can be specified as follows: 1 f(X ) s4 s3 ( f ( X ) b2 ( X ) b2 ) 2 1 ( a1 (8) 2 ( f ( X )) 2 ( b1 ) t3 ( f ( X ) b1 ) (9) (10) where a1 = b1, a3 = b3, t3 >0, s3 > s4 > s5 > 0, 1 ( a1 ) 2 (b1 ) 1 ( a1 ) 0, 1 ( a3 ) 2 (b1) 0, 2 (b3 ) 1 (b2 ) 2 (b2 ) 1 (b2 ) 2 (b2 ) 1 ( a3 ) 1 2 (b3 ) 1 1 (11) (12) ) s3 ( f ( X ) a1 ) 2 s5 s4 ( f ( X ) a2 f ( X ) a2 ) f

354 C.-S. Yu and H.-L. Li and b2 1 1 [ 2 (b2 )] . (13) 1 , 2 1 , 2 1 1 s2 s4 s5 t2 s1 t1 s3 t3 2 2 a1 b1 b2 = a2 b2 1 1 a3 f(X) b3 ( 2 a1 b1 b2 = a2 b2 1 1 a3 ( 2 f(X) b3 (b2 )) 2 (b2 )) 1

(a) Two piecewise linear membership functions 1 and (b) Two piecewise linear membership functions and 2 Figure 7. Membership functions Next, Proposition 2 is presented below: PROPOSITION 2. Function 1 specified in E qs. (9) (13) is a concave function. Proof: Due to s3 > s4 > s5, based on Remark 2, b2 and a2 become concave-type points on 1 . Consequently, 1 is a concave func tion. Consider the following example: Example 2: Maximize Subject to: 2 ( z2 ) , 1 ( z1 ) , z1 = x1 + 2x2, z2 =2x1 + x2, x1 + 3x2 6, x1 + 3x2 12, 4x1 + 3x2 30, 3x1+x2 15, x1 , x 2 0 where 1 ( z1 ) and 2 ( z 2 ) are depicted in Figure 8(a).

Quasi-Concave and Nonconcave FMODM Problem 355 Referring to Remark 1, we know 1 ( z1 8 ) is a convex-type point in 5 vex-type point in z 2 . Then, based on Remark 3, 2 ( z2 1 5 )) = 2.67 pping points can be computed by b1 1 ( 2 ( z2 1 b2 ( 1 ( z1 8 )) = 7. ence to Remark 4, we have two converted functions below: [as shown in )] z 1 and 1 , 2 Z2 1.0 0.2 0.6 0.075 0.2 0 1 2 3 0.1 4 5 6 8 0.2 Z1 9 10 2.67=b1 (a) Two functions , 1 6.4 7= b 2 and 2 in Example 2. 1 2 1.0 0.842104 0.7157907 s2 0.4210536t t2 t3 s3 Z2 Z1 t1 s1 0 1 2 3 4 5 and 6 7 8 9 10 2.67 (b) Two functions 1 2 6.4 in Example 2. Figure 8. Membership functions 1 ( z1 ) = s1(z1 ) is a con and the ma 2 In refer Figure 8(b

0) + s2 2 s1 ( z1 2.67 + z1 2.67) (14) + 2 ( z2 s2 2 s1 3) + ( z1 6.4 + z1 6.4) t 2 t1 ( z2 2 ) =t1(z2 6.4 + z2 6.4) (15) + s2 2 s1 ( z2 7 + z2 7)

356 C.-S. Yu and H.-L. Li 1 where 2 (3) 1 (0) 2 (3) 1 (0) 0 , 0 , 2 (9) 1 1 (10) 1 (10) = 2.67s1 + 3.73s2 + 3.6s3 = 1, 2 (9) =3.4t1 + 0.6t2 + 2t3 = 1, (2.67) 2 (5) 1 (2.67) 2 (5) 2.67 s1 =1 2t1 2.67 s1 3.73s2 =1 3.4t1 1 (6.4) 2 (6.4) 1 1 (6.4) 2 (6.4) (8) 2 (7) (8) 2 (7) 2.67 s1 3.73s2 1.6 s3 =1 3.4t1 0.6t2 s1>s2>s3>0, and t1>t2>t3>0. After computed, the slopes of two converted concave functions are s1 = 0.157698, s2 = 0.079018, s3 = 0.078947, t1 = 0.210526, t2 = 0 .210526, and t3 = 0.078947. Hence, Example 2 can be transformed into Example 2: Maximize subject to x1 + 3x2 6, 1 ( z1 ), 2 ( z2 ) , z1 = x1 + 2x2, z2 = 2x1 + x2, x1+3x2 12, 2 ( z2

4x1 + 3x2 30, 3x1 + x2 15, x1 , x 2 0 where 1 ( z1 ) and respectively. 1 ( z1 ) are expressed in Eqs. (15) and (16), ( z1 2.67 + z1 2.67) + 0.000071 2 ) = 0.157698z1 0.07968 2 ( z1 6.4 + z1 6.4) 2 ( z2 (16) 3) + 0 ( z2 2 6.4 + z2 6.4) + 0 . 131579 2 ) = 0.210526(z2 ( z2 7 (17) Assume that R is the universal set of real numbers, D is an arbitrary domain, an d Rn denotes n-dimensional Euclidean space. For any realvalued function u: D->R, the image of D by u is denoted by u(D); i.e., u(D)={u(d) d D}. Then Inuiguchi et al. (1990) proved that there exists a + z2 7)

Quasi-Concave and Nonconcave FMODM Problem 357 strictly increasing and objective function g: u(D)->u (D) such that u (d) = g(u( d)) for any d belonging to D, where u:D->R and u :D->R. Define an r-level set of u:D->R by [u]r = {d D f(d) r} where r R. Inuiguchi et al. (1990) proved that the solution maximizing a function u is equal to the solution maximizing a funct ion u in any restricted domain when {[u]r r u(D)}={[u ]r r u (D)} and [u ] r is a objective function of [u]r. Accordingly, we have the following proposi tion. PROPOSITION 3. The optimal solution of P1 is the same as that of P2; P1 an d P2 are given below in which 1 , 2 , 1 , and 2 are specified in Eqs. (3) (13): P2 P1 Maximize Maximize Subject to Subject to 1 ( f(X)) 1 ( f(X)) 2 (f(X)) 2 ( f (X)) a1 = b1 f(X) a3 =b3 a1 = b1 f(X) a3 =b3 f(X) F (F is a feasible set). f(X) F (F is a feasible set). Proof. For an f(X) in the restricted domain [a1, a3] or [b1, b3], we have 1(a1) = 1 (a1), 1 (a3 ) 1 1 (a3 ) , 1 2 (b1 ) 1 2 (b1 ) , 1 2 (b3 ) 2 (b3 ) 1 (b2 ) 2 (b2 ) 1 (b2 ) , 2 (b2 ) (a2 ) 2 ( a2 )

(a2 ) , 2 ( a2 ) (b2 ) 2 (b2 ) (b2 ) , 2 (b2 ) and t3 > 0, s3 > s4 > s5 > 0. Since { 1 (f(X)), 2 (f(X))} is the strictly increa sing and bijective function of { 1 (f(X)), 2 (f(X))}, max min{ 1 ( f ( X )), 2 ( f ( X ))} is f(X ) 2( equivalent to max min{ f(X ) 1( f ( X )) , f ( X )) }. Therefore, the optimal solution of P1 is the same as the optimal solution of P2. Take Example 2, for in stance. Solve Example 2 by using Proposition 4, discussed in next paragraph, the obtained solution z1 = 3.525553, z2 = 5.321128, x1 = 1.423341, x2 = 2.474447. T he optimal solution of Example 2 is the same as the optimal solution of Example 2.

358 C.-S. Yu and H.-L. Li PROPOSITION 4. By referring to Proposition 1, consider an FMODM problem as follo ws: Maximize subject to where i ( zi i ( zi ), X F (a feasible set), )= i ( a1 ) +s1( zi ( X ) a1 )+ m 1 sk k 2 sk 2 1 ( z i ( X ) a k z i ( X ) a k ) is

a concave function (i.e., s k sk 1 0 for k = 2, 3, ..., m-1). This FMOP problem can then be reformulated as follows: Maximize subject to i ( zi (17) i ( zi ) m 1 k 2 )= i ( a1 ) + s1( zi ( X ) a1 ) + m 1 ( sk a 1 sk 1

)( zi ( X ) ak k 1 1 d ) zi ( X ) am 1 d 1 0 ,0 d 1 a for all , = 2, 3, ..., m-1, 2 X F (a feasible set). Proof. By referring to Li (1996), a GP problem m 1 k 2 Maximize w = ( zi ( X ) ak m 1 k 2 +zi(X) ak) subject to: zi(X)

0 and 0 < a2 < a3 < … < am-1. is equivalent to Maximize w = 2 ( zi ( X ) ak rk 1 ) subject to: zi(X) - ak + rk-1 0 for k = 2, 3, ... , m-1, rk-1 variables 0, xi 0, where rk-1 are deviation (18) Eq. (20) implies if zi(X) < ak then at optimal solution rk-1 = ak zi(X); if zi(X ) ak then at optimal solution rk-1 = 0. Substitute rk-1 by a 1

k 1 1 d , where d is within the bounds as 0 d a , Equation (20) then becomes

Quasi-Concave and Nonconcave FMODM Problem 359 Maximize w=2 m 1 k 2 ( zi ( X ) a k k 1 1 d ) (19) subject to zi(X) + d1 a3 zi(X) + d1 + d2 a2 zi(X) + d1 + d2 + ... + d m 0 d a 1 1 2 am 1 a for a m 2 = 1, 2, …, m 2 and zi(X) , it is clear that m 3 2 1 0. Since a zi(X) d for all am 1 d am d … a3 d1 d2 a2

d1 0. 1 The first (m-3) constraints in Model (21) therefore are covered by the (m-2)-th constraint in Model (21). Proposition 4 is then proven. Consider the following e xample as depicted in Figure 9(c): Example 3: Maximize z = 1.5x subject to 0.5 ( x 3 2 + x 2) 0.75 ( x 3 + x-3) 2 x Z 5 4 3 2 1 0 2.5. 0.25 1 1.5 d1 1 d2 2 3 4 5 6 7 8 x Figure 9. Z function

360 C.-S. Yu and H.-L. Li Referring to Proposition 4, Example 3 can be linearized as Example 3: Maximize z = 1.5x 0.5(x 2 + d1) 0.75(x 3 + d1 + d2) subject to x + d1 + d2 3, 0 d1 2, 0 d2 1, and x 2.5. After running on LINGO, we obtained z = 3.5, x = 2.5, d1 =0, and d2 = 0.5. 4. SOLUTION ALGORITHMS From the basis of Proposition 1 to Proposition 4, we propose Algorithm 1 for tre ating a quasi-concave FMODM problem. From the basis of Algorithm 1, Algorithm 2 is developed for solving an FMODM problem with more general nonconcave membershi p functions. Algorithm 1 (Solve a quasi-concave FMOP problem): Step 1. Use Propo sition 1 to express each piece-wise membership function as i i ( a i1 ) s ik s i1 ( z i ( X ) s ik 2 a i1 ) ( zi ) M (i ) 1 1 ( z i ( X ) a ik zi ( X ) a ik ) k 2 where aik , k = 1, 2, ..., m are the break points of i ( zi ) , sik , k = 1, 2, ..., m1, are the slopes of line segments between aik and ai ,k 1 , and i =1, 2, ..,n. Step 2. Use Remark 1 to find the convex-type break points and Remark 3 to obtain the corresponding mapping points. Step 3. Use Remark 4 to specify the con verted concave membership functions. Step 4. Use Equations (11) (13) to compute the slopes of the converted concave membership functions. Step 5. Use Propositio n 4 to linearize the converted functions and then solve it by LP techniques. Bas ed on the above discussion, for tackling more general non concave FMODM problems the following remark is presented. Remark 5 (Model the union of some quasi-conc ave membership functions). Any piece-wise membership function can be regarded as the

Quasi-Concave and Nonconcave FMODM Problem 361 union of some quasi-concave membership functions. Take Figure 1(d) for instance, i ( zi ) can be regarded as the union of three quasi-concave functions i 1 ( a1 z i a 2 ) , i 2 ( a2 zi a3 ) , and i 3 ( a3 z i a4 ) . The program of Maximize subject to i ( zi ) for i = 1, 2, …, n can be rewritten as the following program by referring to Li & Yu (1991). Maximi ze Subject to i 3 ( zi i 1 ( zi ) M 1 , 1 i 2 ( zi 2 3 ) M 3, ) M 1, 2 where M is a big number and 1 , 2 , 3 are zero-one variables. From the basis of Remark 5, Algorithm 2 for solving a general non concave FMOP problem is describe d as follows. Algorithm 2 (Solve a general nonconcave FMOP problem): Step 0. Con vert the piece-wise membership functions into the union of some quasi-concave me mbership functions by adding some zero one variables. Steps 1 5 are the same as in Algorithm 1. 5. NUMERICAL EXAMPLES How to solve Example 1 using Algorithm 1 is illustrated below: Step 1 . Use Prop osition 1 to represent following Equations [as depicted in Figure 2(a)]. ( z1 ) 0.04( z1 z1 12) 3) 0.02( z1 2 z1 17) 21 30 z 2 30) 1 ( z1 ) and 2 ( z2 ) as

2) 0.1( z1 12 1 0.04( z1 17 2 ( z2 ) 0.06( z 2 z2 7) 0.02( z 2 17 21) 0.03335( z 2 z 2 17) 0.0665( z 2 27 z 2 27) 0.075( z 2 z1

362 C.-S. Yu and H.-L. Li Step 2. Use Remark 1 to find convex-type points and Remark 3 to calculate their corresponding mapping points as follows [as depicted in Figure 10(a)]: b11 and b22 1 1 [ 2 (17)] 7 , 1 b21 91 . 3 1 2 [ 1 (2)] 31 3 2 [ 1 (17)] 1 ( z1 ) 2 ( z2 ) 1.0 0.8 0.6 0.5 0.4 0.2 0 3 2 7 12 17 21 27 30 32 b11 b 21 b 22 (a) Two quasi-concave membership functions in Example 1. 1 ( z1 ) 2 ( z2 ) 1.0 s2 0.8 0.769 t2 0.5 0.385 1 t3 t4 t5 s3 s1 t3 t 7 12 21 27 32 b 22 =91/3 0

3 b11 =7 b21 =31/3 (b) Two converted concave membership functions in Algorithm 1. Figure 10. Membership functions

Quasi-Concave and Nonconcave FMODM Problem 363 Step 3. Use Remark 4 to specify the converted concave membership functions 1 ( z 1 ) and 2 ( z 2 ) , as shown in Figure 10(b). s1 ( z1 1 ( z1 ) 3) s2 s2 2 s1 ( z1 7 7) (20) s3 2 ( z1 12 t1 ( z2 7) 2 ( z2 ) z2 z2 t2 t1 31 31 t3 t2 ( z2 z2 ) ( z2 21 2 3 3 2 t t t t 21) 4 3 ( z2 27 7) 5 4 ( z2 30 2 2 t 6 t5 91 91 30) ( z2 z2 ) 2 3 3 (21) z2 2 z1 12) z1

Step 4. Use Eqs. (11) (13) to compute the slopes si and t j , i = 1, 2, 3 and j = 1, 2, ..., 6 in (20) and (21). Then 1 (12) (27) (21) 1 (12) 10 s1 5s2 1 0 1 1 (27) 10 s1 5s2 15s3 (21) 26 16 t1 t2 3 3 10 t1 3 32 t2 3 1 2

2 2 (32) 2 (32) 6t3 3t4 1 t5 3 5 t6 3 0 (2) 31 ) 2( 3 1 (2) 31 ) 2( 3 1 5s1 10 t1 3 1 (7) 2 (17) 1 (7) 2 (17) 1 10 s1 10 20 t1 t 3 3 2 1

364 C.-S. Yu and H.-L. Li (14) 2 (27) 1 (14) 2 (27) 1 10 s1 5s2 2 s3 10 32 t1 t2 6t3 3 3 1 (16) 2 (30) 1 (16) 2 (30) 1 10 s1 5s2 4s3 10 32 t1 t2 6t3 3t4 3 3 10 s1 5s2 5s3 10 32 t1 t2 6t3 3t4 3 3 t1 >t2 >t3 >t4 >t5 > t6 1 (17) 91 ) 2( 3 1 (17) 91 ) 2( 3 1 1 t5 3 1 s1 >s2 >s3, After running on the LINGO (2005), the found solutions are s1 = 0.077, s2 = 0.04 6, s3 = 0.067, t1 = 0.11539, t2 = 0.058, t3 = 0.022, t4 = –0.044, t5 = 0.2, and t6 = 0.4. Therefore, we have 1 ( z1 ) 0.077( z1 3) 0.015( z1 7 z1 z1 12)

7) 0.056( z1 12

0.115( z2 7) 0.029( z2 2 ( z2 ) z2 21) 0.011( z2 0.1( z2 31 31 z2 ) 0.0399( z2 21 3 3 27 z2 27) 0.078( z2 30 z2 30)

91 91

z2 ) 3 3

Step 5. Use Proposition 4 to linearize the converted functions and then solve it by linear programming techniques. Based on Proposition 4, the linearized progra m is described below: FMODM Model 7 (Yu and Li Method for Example 1) Maximize su bject to 1 ( Z1 ) 0.067 Z1 0.031d1 0.113d 2 1.804

Quasi-Concave and Nonconcave FMODM Problem 365 2 (Z2 ) 0.4Z 2 0.058d3 0.0798d 4 0.022d5 0.156d 6 0.2d 7 12.808 0, z2 31 d3 3 0 z1 7 d1 z2 z2 0, z1 12 d 2 x1 21 d 4 0, z2 27 d5 0, z2 30 d 6 0 91 d 7 0, z1 x1 2x2 , z2 2 x1 x2 , x1 3 x2 21 3 3x2 27, 4 x1 3 x2 45 , 3x1 x2 30, x1 , x 2 0 By solving on the LINGO, we obtained x1 = 5.62, x 2 = 7.13, z 1 = 8.64, and z 2 = 18.36 in which is exactly the optimal solution of Example 1. Table 1 summarize s the efficiency comparison between Algorithm 1 and conventional FMODM methods f or solving Example 1. Table 1. Efficiency Comparison for Solving Example 1 FMOP Models Required extra Required Required subproblems zero one constraints variables Narasimhan’s and Cann ot treat Example 1 Hannan’s methods FMODM Model 3 (Inuiguchi et al. 3 2 0 Method) FMODM Model 4 (Yang et al. 3 9 0 Method) FMODM Model 6 0 9 8 (Nakamura Method) F MODM Model 7 0 2 0 (Yu and Li Method) Required LP computation Required point cal culation 1 5 1 10 8 0 1 3 Now let us consider the following piece-wise nonconcave FMODM problem.

366 C.-S. Yu and H.-L. Li Example 4 Maximize subject to 0.04( z1 0.02( z 2 3) 1 ( zi ) 0.02( z1 z2 42) 2 z1 2) 0.1( z1 27 12 z1 12) = 0.04( z 17 42 0.04( z1 2 ( z2 )= 0.06( z 2 z2 z2 21) 30) 1 ( zi 7) 0.02( z 2 17 z 2 27 z 2 z 17) 1 1 27) z1

17) 27) 0.0665( z 2 0.075( z 2 21 30 37)

0.03335( z 2 0.145( z 2 2 ( z2 32 32) 0.03( z 2 37 z 2 z 2

where 11(a). 1, 2 ) and ) are non concave functions as depicted in Figure 1.0 0.8 0.6 0.5 0.4 0.2 0 1, 2 1 ( z1 ) 2 (z2 ) z 1 , z2 3 2 7 12 17 21 27 b11 1 ( z1 ) b21 30 32 b22 b12 37 42 45 b 23 17 1.0 s2 0.769 t2 s1 0.385 t1 0 3 2 7 (a) Two non-concave membership functions in Example 4 t8 2 (z2 ) s5 t3 t4 s3 t5 s4 t6 t7 z 1 , z2 12 17 21 27 30 32 37 45 47 (b) Two converted concave membership functions in Alg orithm 2 Figure 11. Membership Function By referring to Algorithm 2, the following steps are illustrated to solve Exampl e 4.

Quasi-Concave and Nonconcave FMODM Problem 367 Step 0. Here 1 ( z1 ) can quasi concave functions 11 ( 2 ( z 2 ) can be regarded as the z 2 32 ) and 22 ( 32 z 2 21 ( 7 be regarded as the union of two 3 z1 27 ) and z1 47 ) . 12 ( 27 union of two qua si concave functions 45 ) . In reference to Remark 1, Example 2 can reformulated as Maximize subject to 12 11 (23) ( 3 z1 27) M 1 1 (27 (7 (32 z1 z2 z2 47) M (1 32) M 2 ) 21 22 45) M (1 1, 2 2 ) where M is a big number and Step 1. 12 are 0-1 variables. 11 (27 Employ Proposition 1 to represent z1 47) , 21 (7 z2 32) and 22 (32 z2 ( 3 z1 27) , 45) as follows: 11 ( 3 z1 27) (27 z1 47)

0.04( z1 3) 0.02( z1 2 z1 2) 0.1( z1 12 0.04( z1 27) 0.02( z1 42 z1 42) 12 0.06(z2 7) 0.02( z2 17 z2 17) 21 (7 z2 32) z2

z1 12) 0.04( z1 17

z1 17)

0.0665( z2 21 z2 21) 0.03335( z2 27 z2 27) 0.075( z2 30 z2 30) 32 ) 0.03( z 2 37 z 2 37 ) 22 ( 32 45 ) 0.04( z 2 Step 2. Based on Remarks 1 and 3, after finding the convex-type point, then the mapping points can be obtained by following equations: b11 1 1 [ 2 (17)] 7 , b12 1 1 [ 2 (37)] 32

368 C.-S. Yu and H.-L. Li b21 1 2 [ 1 (2)] and b23 31 , b22 3 1 41 . 2 [ 1 (42)] 1 2 [ 1 (17)] 91 3 Step 3. Using Remark 4 to specify the converted functions 11 ( z 1 ), 12 ( z 1 ), 21 ( z 2 ), and 22 ( z 2 ) as shown in Figure 11(b), respectively: s1 ( z1 3) 11 ( z1 ) s2 2 s1 ( z1 7 (24) s3 s2 ( z1 12 12 ( z1 ) s4 ( z1 27) s5 2 z1 12) 2 z1 7)

s4 ( z1 32 (25) t1 ( z2 21 ( z2 ) z2 z2 31 31 t3 t2 t2 t1 ( z2 z2 ) ( z2 21 2 3 3 2 t t t t 21) 4 3 ( z2 27 7) 5 4 ( z2 30 2 2 t 6 t5 91 91 30) ( z2 z2 ) 2 3 3 7) (26) 22 ( z2 ) t7 ( z2 32) t8 t7 ( z2 2 41 z2 41) (27) Step 4. In reference to Eqs. (10) 12), the slopes si and t j , i = 1, 2, .., 5 a nd j = 1, 2, ..., 8, in Eqs. (24) (27) can be computed by solving the following equations: s1>s2>s3, s4>s5, t1>t2>t3>t4>t5>t6, t7>t8 (12) 11 (27) 11 11 (12) 10 s1 5s2 1 10 s1 5s2 15s3 11 (27) 0 z2 2 z1 32)

Quasi-Concave and Nonconcave FMODM Problem 12 21 369 (47) (21) (32) 12 21 (47) 5s4 15s5 (21) (32) 1 1 6 t3 3t 4 1 t5 3 5 t6 3 0 21 21 26 t1 3 10 t1 3 16 t2 3 32 t2 3 22 (45) (2) 31 3 22 (45) 9t7 11 21 4t8 5 s1 10 t1 3 1 1 11 21 (2) 31 3 (7) 21 (17) 11 (7) 21 (17) 11 10s1 10 20 t1 t2 3 3 1 (14) 21 (27)

11 (14) 21 (27) 11 10s1 5s2 2s3 10 32 t1 t 6t3 3 3 2 1 (16) 21 (30) 11 (16) 21 (30) 11 10s1 5s2 4s3 10 32 t1 t2 6t3 3t4 3 3 1 11 21 (17) 91 3 11 21 (17) 91 3 10 t1 3 10 s1 5s2 5s3 32 t2 6t3 3t4 3 1 t5 3 1 12 (32) 22 (37) (42) 22 (41) 12 (32) 22 (37) 10 t1 3 10s1 5s2 15s3 5s4 32 1 5 t2 6t3 3t4 t5 t 6 5t 7 3 3 3 1 12 12 (42) 22 (41) 10 s1 5s2 15s3 5s4 10 s5 10 32 1 5 t1 t2 6t3 3t4 t5 t6 9t7 3 3 3 3 1

370 C.-S. Yu and H.-L. Li After computing on the LINGO, the found solutions are s1 = 0.077, s2 = 0.046, s3 = 0.067, s4 = 0.091, s5 = 0.0364, t1 = 0.11539, t2 = 0.058, t3 = 0.022, t4 = 0. 044, t5 = 0.2, t6 = –0.4, t7 = 0.091, and t8 = 0.0455. Therefore, the program (23) becomes Maximize subject to 0.077( Z1 3) 0.0154( Z1 7 1 0.158( Z1 27) 0.0273( Z1 42 1 ) 0.11539(Z2 7) 0.029( Z2 Z2 21) 0.011( Z2 0.1( Z2 91 91 31 31 2 Z2 ) 0.0399( Z2 21 3 3 27 Z2 27) 0.078( Z2 30 Z2 ) M 3 3 Z2 30) Z1 7) 0.056576( Z1 12 Z1 42) M (1 Z1 12) M

0.49( Z 2 32) 0.023( Z 2 41 Z 2 41) M (1 2 ), where M is a big number and 1 , 2 are osition 4, the above problem can then .067Z1 0.031d1 0.113d2 + 1.804 + M 1, 58d4 0.0798d5 0.022d6 0.156d7 0.2d8 + 2), z1 7 d1 0, z1 12 d 2 0, z2 0, z1 42 d3 21 d5 0, z2 0, z2 0 27 d 6 0 0 z2 (31/ 3) d 4 z2 30 d 7 0, z2 (91/ 3) d8 41 d9 zero one variables. Step 5. Employing Prop be linearized below: Maximize ’ Subject to ’ 0 ’ 0.103Z1 0.0556d3 1.973 + M(1 1), ’ 0.4Z2 0.0 12.808 + M 2, ’ 0.444Z2 0.046d9 13.794 + M(1

Quasi-Concave and Nonconcave FMODM Problem 371 z1 x1 2x2 , z2 45 , 3x1 2x1 x2 x2 , x1 3 x2 0 21, x1 3x2 27 4 x1 3x2 30, x1 , x 2 After running on LINGO, the obtained solutions are x1 = 5.62, x 2 = 7.13, z1 = 8 .64 and z 2 = 18.36. This is exactly the optimal solution of Example 2. Table 2 displays the comparisons between traditional FMODM methods and Algorithm 2. Table 2. Efficiency Comparison for Solving Example 2 FMODM models Narasimhan’s, Hannan’s and Inuiguchi et al. methods Yang et al. Method Nakamura Method Yu and Li Method Required zero- Required extra one variables con straints Cannot treat Example 2. Required subproblems Required LP computation Re quired point calculation 9 0 2 26 26 4 0 26 0 1 26 1 14 0 5 6. CONCLUDING REMARKS With the remarkable advance of computer technology in the last two decades, how to solve real-world FMODM problems to obtain the best acceptable solution with a n efficient algorithm has received considerable attention among scientists, engi neers, and managers. Since the powerful advantage of a computerized system stron gly depends on the availability and effectiveness of a mathematical formulation, in reality decision makers widely use either stochastic or fuzzy programming to treat uncertain MODM problems. Stochastic uncertainty is related to environment data such as consumer demand and inflows, whereas fuzzy uncertainty concerns th e use of approximate values by the decision maker when setting objective values (Abdelaziz et al., 2004). Consequently, after Zimmermann first introduced conven tional LP and multi-objective LP into fuzzy set theory, various methods using LP were developed to tackle the FMODM problems.

372 C.-S. Yu and H.-L. Li However, membership functions, whenever used in LP optimization, as reported in literature are generally restricted to linear, triangular, or trapezoid function s. This main restriction has excluded many important domains of application. Man y empirical studies (Biswal, 1997; Hannan, 1981; Mjelde, 1983; Nakamura, 1984; N arasimhan, 1980; Yang et al., 1991;) report that real-world membership functions in the engineering, physical, business, social, and management fields are not p ure linear, triangular, concave, or convex shapes but rather than more general n on concave curves. Therefore, this chapter has been devoted to solving a quasi-c oncave or more general nonconcave FMODM problem. Comparing with conventional FMO DM methods, the proposed method can directly solve a quasi-concave FMODM problem by using standard LP techniques. Moreover, there is no requirement to add extra zero one variables or to divide the original problem into several sub-problems for solving a quasiconcave FMODM problem. Without a tiresome solution process, t he proposed method can be extended to solve more general nonconcave FMOP problem s by adding less number of zero one variables. Numerical examples are employed t o illustrate the practicability and applicability of the proposed method. REFERENCES Abdelaziz, F.B., Enneifar, L., and Martel, J.M., 2004, A multiobjective fuzzy st ochastic program for water resources optimization: The case of lake management, INFOR, 42: 201 215. Biswal, M.P., 1997, Use of projective and scaling algorithm to solve multi-objective fuzzy linear programming problems, The Journal of Fuzzy Mathematics, 5: 439 448. Hannan, E.L., 1981a, Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems, 6: 235 248. Hannan, E.L., 1981b, On fuzzy g oal programming, Decision Sciences, 12: 522 531. Inuiguchi, M., Ichihashi, H., a nd Kume, Y., 1990, A solution algorithm for fuzzy linear programming with piecew ise linear membership functions, Fuzzy Sets and Systems, 34: 15 31. Lai, Y.J., a nd Hwang, C.L., 1994, Fuzzy Multiple Objective Decision Making, SpringerVerlag, New York. Li, H.L., 1996, Technical note: An efficient method for solving linear goal programming problems, Journal of Optimization Theory and Applications, 90: 465 469. Li, H.L., and Yu, C.S., 1999, Comments on “fuzzy programming with nonlin ear membership functions …,” Fuzzy Sets and Systems, 101: 109 113. Mjelde, K.M., 198 3, Fractional resource allocation with S-shaped return functions, Journal of Ope rational Research Society, 34(7): 627 632. Nakamura, K., 1984, Some extensions o f fuzzy linear programming, Fuzzy Sets and Systems, 14: 211 229.

Quasi-Concave and Nonconcave FMODM Problem 373 Narasimhan, R., 1980, Goal programming in a fuzzy environment, Decision Sciences , 11: 325 336. Romero, C., 1994, Handbook of Critical Issues in Goal Programming , Pergamon Press, New York. LINGO 9.0, 2005, LINDO System Inc., Chicago. Simon, H.A., 1960, Some further notes on a class of skew distribution functions, Inform ation and Control, 3: 80–88. Yang, T., Ignizio, J.P., and Kim, H.J., 1991, Fuzzy p rogramming with nonlinear membership functions: piecewise linear approximation, Fuzzy Sets and Systems, 41: 39 53. Yu, C.S., and Li, H.L., 2000, Method for solv ing quasi-concave and non-concave fuzzy multi-objective programming problems, Fu zzy Sets and Systems, 109: 59–82. Yu, C.S., 2001, A Method For Solving Quasi-Conca ve Or General Non-Concave Fuzzy Multi-Objective Programming Problems And Its App lications In Logistic Management, Marketing Strategies, And Investment DecisionMaking, National Science Council of R.O.C., Taipei. Zimmermann, H.J., 1976, Desc ription and optimization of fuzzy systems, International Journal of General Syst ems, 2: 209 215. Zimmermann, H.J., 1981, Fuzzy programming and linear programmin g with several objective functions, Fuzzy Sets and Systems, 1: 45 55.

INTERACTIVE FUZZY MULTI-OBJECTIVE STOCHASTIC LINEAR PROGRAMMING Masatoshi Sakawa1 and Kosuke Kato2 1 Department of Artificial Complex Systems Engineering, Graduate School of Enginee ring, Hiroshima University 2Department of Artificial Complex Systems Engineering , Graduate School of Engineering, Hiroshima University Abstract: Two major approaches to deal with randomness or ambiguity involved in mathematic al programming problems have been developed. They are stochastic programming app roaches and fuzzy programming approaches. In this chapter, we focus on multiobje ctive linear programming problems with random variable coefficients in objective functions and/or constraints. Using several stochastic models such as an expect ation optimization model, a variance minimization model, a probability maximizat ion model, and a fractile criterion optimization model in chance constrained pro gramming, the stochastic programming problems are transformed into deterministic ones. As a fusion of stochastic approaches and fuzzy ones, after determining th e fuzzy goals of the decision maker, several interactive fuzzy satisfying method s to derive a satisfying solution for the decision maker by updating the referen ce membership levels are presented. Fuzzy mathematical programming, multi-criter ia analysis, linear programming, stochastic programming, interactive programming Key words: 1. INTRODUCTION In actual decision-making situations, we must often make a decision on the basis of vague information or uncertain data. For such decision-making problems invol ving uncertainty, there exist two typical approaches: stochastic programming and fuzzy programming. C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 375

376 M. Sakawa and K. Kato Stochastic programming, as an optimization method on the basis of the probabilit y theory, has been developing in various ways (Stancu-Minasian, 1984; 1990), inc luding a two-stage problem by G.B. Dantzig (1955), and chance constrained progra mming by A. Charnes and W.W. Cooper (1959). In particular, for multi-objective s tochastic linear programming problems, I.M. Stancu-Minasian (1984, 1990) conside red the minimum risk approach, while J.P. Leclercq (1982) and Teghem Jr. et al. (1986) proposed interactive methods. On the other hand, fuzzy mathematical progr amming representing the vagueness in decision-making situations by fuzzy concept s has been studied by many researchers (Lai and Hwang, 1992; Rommelfanger, 1996; Sakawa, 1993). Fuzzy multiobjective linear programming, first proposed by H.-J. Zimmermann (1978), has also been developed by numerous researchers, and an incr easing number of successful applications have been introduced (Delgado et al., 1 994; Kacprzyk and Orlovski, 1987; Lai and Hwang, 1994; Luhandjula, 1987; Sakawa et al. 1987; Sakawa, 2001; 1993; 2000; Slowinski, 1998; Slowinski and Teghem, 19 90; Verdegay and Delgado, 1989; Zimmermann, 1987). As a hybrid of the stochastic approach and the fuzzy one, Wang et al. (Wang and Qiao, 1993) and Luhandjula et al. (Luhandjula, 1996; Luhandjula and Gupta, 1996) considered mathematical prog ramming problems with fuzzy random variables (Kwakernaak, 1778; Puri, 1986), and Liu and Iwamura (1998) discussed chance constrained programming involving fuzzy parameters. In particular, Hulsurkar et al. (1997) applied fuzzy programming to multi-objective stochastic linear programming problems. Unfortunately, however, in their method, since membership functions for the objective functions are sup posed to be aggregated by a minimum operator or a product operator, optimal solu tions that sufficiently reflect the decision maker’s preference may not be obtaine d. Under these circumstances, in this chapter, we focus on multiobjective linear programming problems with random variable coefficients in objective functions a nd/or constraints. Through the use of several stochastic models, including an ex pectation optimization model, a variance minimization model, a probability maxim ization model, and a fractile criterion optimization model together with chance constrained programming techniques, the stochastic programming problems are tran sformed into deterministic ones. Assuming that the decision maker has a fuzzy go al for each objective function, having determined the fuzzy goals of the decisio n maker, we present several interactive fuzzy satisfying methods to derive a sat isfying solution for the decision maker by updating the reference membership lev els. As an illustrative numerical example, a

Fuzzy Muti-objective Stochastic Linear Programming 377 multi-objective linear programming problem involving random variable coefficient s for the probability maximization model is provided to demonstrate the feasibil ity of the proposed method. 2. MULTI-OBJECTIVE LINEAR PROGRAMMING PROBLEMS WITH RANDOM VARIABLE COEFFICIENTS Throughout this chapter, we deal with multi objective linear programming problem s where coefficients in objective functions and right-hand side constants of con straints are assumed to be random. Such multi-objective linear programming probl ems involving random variable coefficients are formally formulated as: Minimize z1 (x, ) c1 ( )x Minimize zk (x, ) ck ( )x subject to A x b( ) x 0 (1) where x is an n-dimensional decision variable column vector and A is an m n coef ficient matrix. It should be noted that cl ( ) , l 1,..., k are n-dimensional ra ndom variable row vectors with finite mean cl and finite covariance matrix l Vl ( v jh ) (Cov[clj ( ), clh ( )]) , j 1,..., n, h 1,..., n and bi ( ) , i 1,..., m are random variables with finite mean bi , which are independent of each other , and the distribution function of each of them is also assumed to be continuous and increasing. Multi-objective linear programming problems with random variabl e coefficients are said to be multi-objective stochastic linear programming ones , which are often seen in actual decision-making situations. For example, consid er a production planning problem to optimize the gross profit and production cos t simultaneously under the condition that unit profits of the products, unit pro duction costs of them, and the maximal amounts of the resources depend on season al factors or market prices. Such a production planning problem can be formulate d as a multi-

378 M. Sakawa and K. Kato objective programming problem with random variable coefficients expressed by Eq. (1). Since the formulated problem Eq.(1) contains random variable coefficients, definitions and solution methods for ordinary mathematical programming problems cannot be directly applied. Consequently, we deal with the constraints in Eq. ( 1) as chance constrained conditions (Charnes and Cooper, 1959), which mean that the constraints need to be satisfied with a certain probability (satisfying leve l) and over. Namely, replacing the constraints in Eq. (1) by chance constrained conditions with satisfying levels i , i 1,..., m , Eq. (1) can be converted as Minimize Minimize subject to Pr a1 x Pr a m x z1 (x , ) z k (x, ) b1 ( ) bm ( ) x c1 ( ) x c k ( )x 1 (2) m 0 where ai is the ith ting continuous and , i 1,..., m by Fi written as: Pr ai x bi ( ) 1 Pr ) i (3) ˆ Letting bi Fi 1 (1 equivalent problem: i) , Eq.(2) can be transformed into the following Minimize z1 (x, ) Minimize z k (x, ) subject to Ax x c1 ( )x ck ( )x ˆ b 0 (4) row vector of A and bi ( ) is the ith element of b( ) . Deno increasing distribution functions of random variables bi ( ) ( r ) Pr[bi ( ) r ] , the ith constraint in Eq.(2) can be re bi ( ) ai x i 1 Fi ( ai x ) i Fi ( ai x ) 1 i 1 ai x Fi (1 i

Fuzzy Muti-objective Stochastic Linear Programming 379 ˆ ˆ ˆ where b (b1 ,..., bm )T . In the following section, for notational convenience, the feasible region of Eq. (4) is denoted by X. For the multi-objective chance c onstrained programming problem Eq. (4), several stochastic models such as an exp ectation optimization model, a variance minimization model, a probability maximi zation model, and a fractile criterion model have been proposed depending on the concern of the decision maker. 3. EXPECTATION OPTIMIZATION MODEL In this section, we state the expectation optimization model for multiobjective chance constrained programming problems (Sakawa and Kato, 2002; Sakawa et al. 20 03b), where the decision maker aims to optimize the expectation of each objectiv e function represented as a random variable in Eq. (4). Substituting the objecti ve functions zl ( x , ) cl ( ) x , l 1,..., k in Eq. (4) for their expectations, the problem can be converted as M inimize M inimize subject to E z1 (x , ) E z k (x , ) ˆ Ax b x 0 (5) Letting cl E[cl ( )] , zl ( x ) E[ zl ( x, )] can be expressed as zl ( x ) cl x . (6) Then, Eq. (5) can be reduced to the following ordinary multi-objective linear pr ogramming problem: Minimize Minimize subject to c1 x ck x ˆ Ax b x 0. (7)

380 M. Sakawa and K. Kato In order to consider the imprecise nature of the decision maker’s judgments for ea ch objective function zl ( x ) cl x in Eq. (7), if we introduce the fuzzy goals such as “ zl ( x ) should be substanti ally less than or equal to a certain value,” Eq. (7) can be rewritten as Maximize ( x X 1 ( z 1 ( x )),… , k ( z k ( x ))) (8) where l ( ) is a membership function to quantify a fuzzy goal for the lth object ive function in Eq. (7). To be more specific, if the decision maker feels that z l ( x ) should be greater than or equal to at least zl ,0 and that z l ( x ) z l ,1 ( z l ,0 ) is satisfactory, the shape of a typical membership function is sh own in Figure 1. Figure 1. An example of a membership function l ( zl ( x )) Since Eq. (8) is regarded as a fuzzy multi-objective decision-making problem, th ere rarely exists a complete optimal solution that simultaneously optimizes all objective functions. As a reasonable solution concept for the fuzzy multi-object ive decision-making problem, M. Sakawa et al. (Sakawa and Yano, 1985; 1990; Saka wa et al., 1987; Sakawa, 1993) defined M-Pareto optimality on the basis of membe rship function values by directly extending the Pareto optimality in the ordinar y multi-objective programming problem.

Fuzzy Muti-objective Stochastic Linear Programming DEFINITION 1 (M-PARETO OPTIMAL SOLUTION) 381 x* X is said to ist another x X j ( x )) j ( z on function D ( written as: be an M-Pareto optimal solution if and only if there does not ex such that l ( z l ( x )) l ( z l ( x*)) for l 1,..., k and j ( z j ( x* )) for at least one j { 1,… ,k } . Introducing an aggregati x ) for k membership functions in Eq. (8), the problem can be re

Minimize subject to x D ( x) X (9) The aggregation function D ( x ) represents the degree of satisfaction or prefer ence of the decision maker for the whole of k fuzzy goals. Following the convent ional fuzzy approaches, as aggregation functions, Hulsurkar et al. (1997) adopte d the minimum operator of Bellman and Zadeh (1970) defined by D (x) l 1,… , k min { l ( z l ( x ))} . and the product operator of Zimmermann (1978) defined by D ( x) k l 1 l ( z l ( x )) . However, it should be emphasized here that such approaches are preferable only w hen the decision maker feels that the minimum operator or the product operator i s appropriate. In other words, in general decision situations, the decision make r does not always use the minimum operator or the product operator when combinin g the fuzzy goals. Probably the most crucial problem in Eq. (9) is the identific ation of an appropriate aggregation function that well represents the decision m aker s fuzzy preferences. If D ( x ) can be explicitly identified, then Eq. (9) reduces to a standard mathematical programming problem. However, this rarely hap pens, and as an alternative, an interaction with the decision maker is necessary for finding a satisfying solution to Eq. (9). In an interactive fuzzy satisfyin g method, to generate a candidate for a satisfying solution that is also M-Paret o optimal, the decision maker is asked to specify the aspiration levels of achie vement for the membership values of all membership functions, called the referen ce membership levels (Sakawa and Yano, 1985; 1989; 1990; Sakawa et al., 1987; Sa kawa, 1993).

382 M. Sakawa and K. Kato For the decision maker s reference membership levels l , l 1,… ,k , the correspond ing M-Pareto optimal solution, which is nearest to the requirements in the minim ax sense or better than that if the reference membership levels are attainable, is obtained by solving the following minimax problem: Minimize max { l 1,…,k l l ( z l ( x ))} . (10) subject to x X By introducing the auxiliary variable v, this problem can be equivalently transf ormed as Minimize v subject to x 1 k 1 ( z1 ( x )) k ( z k ( x )) v v . (11) X If the value of v is fixed to v*, Eq. (11) can be reduced to a linear programmin g problem. Therefore, we can find an optimal solution ( x*, v*) corresponding to v* by the bisection method based on the simplex method. Following the preceding discussions, we can now construct the interactive algorithm in order to derive the satisfying solution for the decision maker from the M-Pareto optimal solutio n set. The steps marked with an asterisk involve interaction with the decision m aker. Interactive fuzzy satisfying method for expectation optimization model Ste p 1. Calculate the individual minimum zlmin and maximum zlmax of E [ zl ( x , )] z l ( x ) , l 1,..., k under the chance constrained conditions with satisfying levels i , i 1, … , m by solving the following linear programming problems: minimize zl ( x ) x X cl x , l 1, … , k cl x, l 1, … , k (12) (13) maximize zl ( x ) x X

Fuzzy Muti-objective Stochastic Linear Programming 383 Step 2. Ask the decision maker to determine membership functions l ( z l ( x )) for objective functions in Eq. (7). Step 3. Ask the decision maker to set the in itial reference membership levels l 1, l 1,… ,k . Step 4. Solve the following mini max problem Minimize max { l 1,…,k l l ( z l ( x ))} (14) subject to x X corresponding to the reference membership levels l , l 1, … , k . To be more speci fic, after calculating the optimal value v* to the problem Minimize v subject to z1 ( x ) z k ( x) x X 1 1 ( ( 1 k v) v) (15) 1 k by the bisection method and phase one of the two-phase simplex method, solve the linear programming problem Minimize z1 ( x ) subject to z 2 ( x ) zk ( x) x X 2 1 1 k ( ( 2 k v* ) v *) (16) where z1 ( x , ) is supposed to be the most important to the decision maker. For the obtained x*, if there are inactive constraints in the first ( k 1 ) constra ints, replace l for inactive constraints with l ( zl ( x*)) v * and resolve the corresponding problem. Furthermore, if the obtained x* is not unique, perform th e M-Pareto optimality test. Step 5. The decision maker is supplied with the corr esponding MPareto optimal solution and the trade-off rates between the membershi p functions. If the decision maker is satisfied with the current membership func tion values of the M-Pareto optimal solution, stop. Otherwise, ask the decision maker to update the reference membership levels l , l 1,..., k by

384 M. Sakawa and K. Kato l considering the current membership function values with the trade-off rates 1 l zl x* together , l 2 ,..., k and return to step 4. Here, the trade-off rates are expressed as 1 ( z1 ( x )) l ( zl ( x )) l 1 ( z1 ( x*)) l ( zl ( x*)) , l 2, … , k where l , l 2, … , k are simplex multipliers in Eq. (16). 2, … , k in step 5 indicat e the Since the trade-off rates 1/ l,l decrement of value of a membership functi on 1 with a unit increment of value of a membership function l , they are employ ed to estimate the local shape of ( 1 ( z1 ( x*)),… , k ( z k ( x ))) around x*. H ere it should be stressed to the decision maker that any improvement of one memb ership function can be achieved only at the expense of at least one of the other membership functions. For further details along this line, the readers might re fer to the corresponding papers (Sakawa et al., 2000; 2003b). 4. VARIANCE MINIMIZATION MODEL Since objective functions regarded as random variables in Eq. (4) are reduced to their expectations in the expectation optimization model, the requirement of th e decision maker for risk is not reflected in the obtained solution. From this v iewpoint, in this section, we consider the variance minimization model for multi -objective chance constrained programming problems (Sakawa et al., 2002). In the model, we substitute the minimization of variances of objective functions for t he minimization of objective functions in Eq. (4). Then, the problem can be rewr itten as Minimize Minimize subject to z1 (x ) z k (x ) A x x V a r z1 (x , V a r z k (x , ˆ b 0 ) ) x T V1 x (17) x T Vk x Using the variance minimization model, the obtained solution might be too bad in the sense of the expectation of objective functions, while it

Fuzzy Muti-objective Stochastic Linear Programming 385 accomplishes the minimization in the sense of the variance. In order to take the requirement of the decision maker for expectations of objective functions into account, we consider the following revised variance minimization model incorpora ting constraints that the expectation of each objective function, zl ( x ) cl x must be less than or equal to a certain permissible level l , l 1, … , k . Minimize Minimize subject to z1 (x ) z k (x ) Ax Cx x 0 Var z1 (x , ) Var z k (x , ) ˆ b x T V1 x x T Vk x (18) where C (c1T ,…, ckT )T and ( 1 ,…, k )T , and we denote the feasible region of Eq. (18) by X . In order to consider the imprecise nature of the decision maker s ju dgments for each objective function in Eq. (18), if we introduce the fuzzy goals such as “ zl (x ) should be substantially less than or equal to a certain value,” t he problem Eq. (18) can be rewritten as Maximize ( 1 ( z1 (x )),… , x X k ( z k (x ))) (19) where l ( ) is a membership function to quantify a fuzzy goal for the lth object ive function in Eq. (18) shown in Figure 2. Figure 2. An example of a membership function l ( zl ( x ))

386 M. Sakawa and K. Kato In order to derive a satisfying solution for the decision maker from the M-Paret o optimal solution set, Sakawa et al. (Sakawa and Yano, 1985; 1990; Sakawa et al ., 1987; Sakawa, 1993) proposed an interactive fuzzy satisfying method such that the decision maker interactively updates the aspiration levels of achievement f or the membership values of all membership functions, called the reference membe rship levels, until he is satisfied. We now summarize the interactive algorithm. Interactive fuzzy satisfying method for variance minimization model Step 1. Ask the decision maker to specify the satisfying levels i 1,… , m for each of the constraints in Eq. (1). zlmax i, Step 2. After calculating the individual minimum zlmin and maximum of E[ zl ( x , )] zl ( x ), l 1, … , k under the chance constrained conditions, ask the decisio n maker to specify permissible levels l , l 1,… ,k for objective functions. Step 3 . Calculate the individual minimum zl ,min of zl ( x ), l 1,… ,k in Eq. (18) by so lving the following quadratic programming problems: T minimize z l ( x ) x X x Vl x , l 1,…, k (20) Step 4. Ask the decision maker to determine membership functions l ( z l ( x )) for objective functions in (18) on the basis of individual minima zl ,min . Step 5. Ask the decision maker to set the initial reference membership levels l 1, l 1, … , k . Step 6. Calculate the optimal solution x* to the augmented minimax problem Eq. ( 21) corresponding to the current reference membership levels l , l 1, … , k . Minimize max x X l l ( z l ( x )) l 1,…,k ( i 1,…,k i i ( z i ( x ))) (21) Here, assuming that each membership functions l ( ) , l 1,… ,k is nonincreasing an d concave, Eq. (21) is a convex programming problem.

Fuzzy Muti-objective Stochastic Linear Programming 387 Under the assumption, we can solve Eq. (21) by a traditional convex programming technique as the sequential quadratic programming method. Step 7. The decision m aker is supplied with the obtained solution x*. If the decision maker is satisfi ed with the current membership function values of x*, stop. Otherwise, ask the d ecision maker to update the reference membership levels l , l 1,… ,k by considerin g the current membership function values l ( zl ( x*)) , and return to step 6. 5. PROBABILITY MAXIMIZATION MODEL In this section, we investigate the probability maximization model for a multi-o bjective chance constrained programming problem (Sakawa and Kato, 2002; Sakawa e t al., 2004), where the decision maker aims to maximize the probability that eac h objective function represented as a random variable is less than or equal to a certain permissible level in Eq. (4). Substituting the minimization of the obje ctive functions z l ( x , ) cl ( ) x , l 1,… ,k in Eq. (4) for the maximization of the probability that each objective function zl ( x , ) is less than or equal t o a certain permissible level f l , the problem can be converted as Maximize p1 ( x ) Maximize p k ( x ) subject to Ax x Pr z1 ( x , ) Pr z k ( x , ) ˆ b 0 f1 fk (22) In order to consider the imprecise nature of the decision maker s judgment for e ach objective function in Eq. (22), if we introduce the fuzzy goals such as “ pl ( x ) should be substantially greater than or equal to a certain value,” Eq. (22) c an be rewritten as M axim ize ( x X 1 ( p1 (x)),… , k ( p k ( x ))) (23) where l ( ) is a membership function to quantify a fuzzy goal for the l th objec tive function in Eq. (22). To be more explicit, if the decision maker feels that pl ( x ) should be greater than or equal to at least pl ,0 and pl ( x ) pl ,1 ( pl ,0 ) is satisfactory, the shape of a typical membership function is shown in Figure 3.

388 M. Sakawa and K. Kato Figure 3. An example of a membership function l ( pl ( x )) In an interactive fuzzy satisfying method, to generate a candidate for the satis fying solution that is also M-Pareto optimal, the decision maker is asked to spe cify the aspiration levels of achievement for the membership values of all membe rship functions, called the reference membership levels (Sakawa and Yano, 1985; 1989; 1990; Sakawa et al., 1987; Sakawa, 1993). For the decision maker’s reference membership levels l , l 1, … , k , the corresponding M-Pareto optimal solution, w hich is nearest to the requirements in the minimax sense or better than that if the reference membership levels are attainable, is obtained by solving the follo wing minimax problem: Minimize max { l 1,…,k l l ( pl ( x ))} . (24) subject to x X. By introducing the auxiliary variable v, this problem can be equivalently transf ormed as Minimize v subject to x 1 k 1 ( p1 ( x )) k v v . (25) X. ( p k ( x )) Now, let every membership function l ( ) be continuous and strictly increasing. Then, (25) is equivalent to the following problem:

Fuzzy Muti-objective Stochastic Linear Programming Minimize v 1 389 subject to p1 ( x ) pk ( x) x X 1 1 k ( ( 1 v) v) . (26) k Since Eq. (26) is a nonconvex, nonlinear programming problem in general, it is d ifficult to solve it. Here, in Eq. (1), we assume that cl ( ), l 1,… ,k dimensiona l random variable row vectors expressed by cl ( ) cl1 t l ( )cl2 where tl ( ) ’s a re random variables independent of each other, and l ( ) ’s are random variables e xpressed by l ( ) l1 tl ( ) l2 , where the corresponding distribution function T l ( ) of each of tl ( ), s is assumed to be continuous and strictly increasing. Supposing that cl2 x l2 0, l 1, … , k for any x X , from the assumption on distrib ution functions Tl ( ) of random variables tl ( ) , we can rewrite the objective functions in Eq. (22) as Pr z l ( x , ) fl Pr ( c1 l Pr ( c l2 x Pr t l ( ) Tl fl c1 x l c l2 x tl ( ) c l2 ) x 2 l ( 1 l tl ( ) 1 l 2 l ) f1 fl )tl ( ) ( c1 x l (c l2 x 1 l ( c1 x l 1 l 2 l ) fl ) )

. 2 l Hence, Eq. (22) can be transformed into the following ordinary multiobjective pr ogramming problem: Maximize p1 ( x ) T1 fl c1 x 1 2 c1 x 2 1 1 1 . Maximize p k ( x ) subject to x X Tk fk 2 k (27) c x c x 2 k 1 k 1 k

390 M. Sakawa and K. Kato In view of fl 2 pl ( x ) Tl cl x cl x 2 l 1 1 l and the continuity and strictly increasing property of the distribution function Tl ( ) , This problem can be equivalently transformed as Minimize v subject to 1 f1 c1 x 2 c1 x 1 1 2 1 T1 1 ( 1 1 ( 1 v )) . fk c1 x k c x 2 k 1 k 2 k (28) Tk 1 ( 1 k ( k v )) x X It is important to note here that, in this formulation, if the value of v +is fi xed, it can be reduced to a set of linear inequalities. Obtaining the optimal so lution v* to the above problem is equivalent to determining the maximum value of v so that there exists an admissible set satisfying the constraints of equation s (28). Since v satisfies max l 1 ,…,k max

l ,max v max l 1 ,…,k min l ,min where max l 1 ,…,k max l , l ,max max x X l( pl ( x )), l ,min min x X l( pl ( x )) we can obtain the minimum value of v by combined use of the bisection method and phase one of linear programming technique. After calculating v*, the minimum va lue of v, we solve the following linear fractional programming problem in order to uniquely determine x* corresponding to v*:

Fuzzy Muti-objective Stochastic Linear Programming Minimize subject to 1 c1 x c12 x 1 1 2 1 1 2 2 2 391 f1 T2 1 ( 1 2 f 2 c1 x 2 2 c2 x fk c x 2 ck x x X 1 k ( 2 v *)) (29) 1 k 2 k Tk 1 ( 1 k ( k v *)) where z1 ( x , ) is supposed to be the most important to the decision maker. Usi ng the Charnes Cooper transformation (Charnes and Cooper, 1962) s 2 1/( c1 x 2 1 ), y s x, s 0 (30) the linear fractional programming problem Eq. (29) is converted to the following linear programming problem 1 Minimize c1 y ( 2 (c2 y 1 1 f1 ) s 2 2 subject to 2

s) s) c1 y 2 c1 y k ( ( 1 2 f2 ) s fk ) s 0 0 k 2 (ck y 2 k 1 k Ay c12 y s y s 0 0 ˆ sb 2 1 0 s 1 (31) is sufficiently small and positive. where l Tl 1 ( 1 l ( l v *)) , and

392 M. Sakawa and K. Kato If the optimal solution ( y* , s * ) to Eq. (31) is not unique, the Pareto optim ality of x* y * s* is not guaranteed. The Pareto optimality of x* can be tested by solving the following linear programming problem. k Maximize w l 1 l subject to q1 (x ) 1 q1 (x*) r1 (x ) r1 (x*) q k (x*) rk (x ) rk (x*) ( 1 ,… , k ) T 0 (32) q k (x ) x X, k where ql ( x ) f 1 c1 x l 1 l , rl ( x ) cl2 x 2 l For the optimal solution to Eq. (32), (a) if w l 1,..., k , x* is Pareto optimal. On the other hand, (b) w 0 , i.e., l 0 for 0 , i.e., l 0 for at least one l, x* is not Pareto optimal. Then, we can find a Pareto optimal solution according to the following algorithm. Step 1. For the optimal solution x , to the problem (32), after arbitrarily selecting j such as j 0 , solve the following problem: Maximize f j c1j x c2j x fl c1x l cl2 x 1 l 2 l 2 j 1 j subject to 1 l 2 l fl c1x l cl2 x 1 l 2 l 1 l 2 l , {l

l 0} (33) 0} fl c1x l cl2 x x X fl c1x l cl2 x , {l l

Fuzzy Muti-objective Stochastic Linear Programming 393 Since the above problem can be converted to a linear programming problem by the Charnes and Cooper transformation (Charnes and Cooper, 1962), we can solve it by the simplex method. x Step 2. To test the Pareto optimality of the optimal solu tion ˆ to Eq. x (33), solve the problem Eq. (32) where ˆ is substituted for x*. x St ep 3. If w 0 , ˆ is Pareto optimal and stop. Otherwise, i.e., if w 0 , x return to step 1 since ˆ is not Pareto optimal. Repeating this process at least k 1 iterati ons, a Pareto optimal solution can be obtained. The decision maker must either b e satisfied with the current Pareto optimal solution or act on this solution by updating the reference membership levels. In order to help the decision maker ex press a degree of preference, trade-off information between a standing membershi p function and each of the other membership functions is very useful. Such trade -off information is easily obtainable since it is closely related to the simplex multipliers of Eq. (31). To derive the trade-off information, define the Lagran ge function L for Eq. (31) as follows: L( y, s, , , ) c1 y ( 1 1 1 1 f1 ) s 1 l 2 k l 2 l[ l m i 1 (cl y i ( ai y 2 2 l s) {cl y ( m 1 n j 1 j f l ) s}] 2 1 ˆ s bi ) (34) m 2 (c1 y yj s 1) s ( s ) n 1 where ,

, and are simplex multipliers. l Then, the partial derivative of L ( y , s , , , ) with respect to as follows. L( y , s, , , ) l l is given ( cl2 y 2 l s ), l 2, … , k (35)

394 M. Sakawa and K. Kato On the other hand, for the optimal solution ( y*, s *) to Eq. (31) and the corre sponding simplex multipliers ( *, *, *) , the following equation holds from the Kuhn-Tucker necessity theorem (Sakawa, 1993): L ( y* , s * , * , * , * ) 1 c1 y * ( 1 1 f 1 ) s* l (36) is calculated as If the first ( k 1) constraints to Eq. (31) are active, follows: cl y * ( l 1 1 l 2 l f l ) s* s* c l2 y * , l 2, … , k . (37) From Eq. (35), Eq. (36) and Eq. (37), for l ( c1 y * ( 1 1 1 2 ,… , k , we have 2 2 l f1 ) s * ) s* 1 1 cl y * ( l f l ) 2 2 cl y * l s* l* ( cl y * s*) . (38) By substituting x* for y*, s* in Eq. (38), the equation is rewritten as 1 f 1 c1 x * 2 c1 x * 2 1 1 l 2 l 1 1 * l cl x * 2 c1 x * 2 fl 2 cl1 x * cl x *

2 l 2 1 ,l 2 ,… ,k . (39) Using the chain rule, the following relation holds: T1 1 f 1 c1 x * 2 c1 x * 2 1 1 l 2 l 1 1 l* cl2 x * 2 c1 x * Tl f l cl1 x * cl2 x * 2 l 2 1 T1 f 1 c1 x * 2 c1 x * 2 1 1 1 1 1 Tl f l cl x * 2 cl2 x * l l 2,… , k . 1 l . (40)

Fuzzy Muti-objective Stochastic Linear Programming 395 Equivalently, p1 ( x*) pl ( x*) l* cl x * 2 c1 x * 2 2 l 2 1 p1 ( x*) , l pl ( x*) 2,…, k . (41) Again, using the chain rule, for l 1 ( p1 ( x*)) l ( pl ( x*)) 2 ,… , k we have 2 l 2 1 l* cl x * 2 c1 x * 2 p1 ( x*) pl ( x*) 1 ( p1 ( x*)) l ( pl ( x*)) . (42) It should be stressed here that in order to obtain the trade-off information fro m Eq. (42), the first ( k 1 ) constraints in Eq. (31) must be active. Therefore, if there are inactive constraints, it is necessary to replace for inactive cons traints with and solve the l l ( pl ( x*)) v * corresponding problem to obtain t he simplex multipliers. Following the preceding discussions, we can now construc t the interactive algorithm in order to derive the satisfying solution for the d ecision maker from the Pareto optimal solution set. Interactive fuzzy satisfying method for probability maximization model Step 1. Calculating the individual mi nimum zlmin and maximum zlmax of E[ zl ( x, )] zl ( x ), l 1, … , k under the chan ce constrained conditions with satisfying levels i , i 1, … , m . Step 2. Ask the decision maker to specify permissible levels f l , l 1, …, k for objective functio ns. Step 3. Calculate the individual minimum pl ,min and maximum pl ,max of pl ( x ), l 1,… , k in the multi-objective probability maximization problem Eq. (27) b y solving the following problems: fl 2 Minimize pl ( x ) Tl

x X cl x cl x 2 l 1 1 l , l 1,… , k (43) Maximize pl ( x ) Tl x X fl 2 cl x cl x 2 l 1 1 l , l 1,… , k (44)

396 M. Sakawa and K. Kato Then ask the decision maker to determine membership functions l ( pl ( x )) for objective functions in Eq. (27). Step 4. Ask the decision maker to set the initi al reference membership levels l 1, l 1,… ,k Step 5. In order to obtain the optimal solution x* to the minimax problem Eq. (2 4) corresponding to the reference membership levels 1, l 1,… ,k , after solving Eq . (28) by the bisection method and phase l one of the two-phase simplex method, solve the linear programming problem Eq. (31). For the obtained x*, if there are inactive constraints in the first ( k 1 ) constraints, replace l for inactive c onstraints with l ( pl ( x*)) v * and resolve the corresponding problem. Further more, if the obtained x* is not unique, perform the Pareto optimality test. Step 6. The decision maker is supplied with the corresponding Pareto optimal solutio n and the trade-off rates between the membership functions. If the decision make r is satisfied with the current membership function values of the Pareto optimal solution, stop. Otherwise, ask the decision maker to update the reference membe rship levels l 1, l 1,… ,k by considering the current membership function values l ( pl ( x*)) together 2 ,… , k , and return to step 5. with the trade-off rates 1/ l, l / l , l 2 ,… , k in Step 6 indicate the Since the trade-off rates 1 decremen t of value of a membership function 1 with a unit increment of value of a member ship function l , they are employed to estimate the local shape of 1 ( p1 ( x*)) ,… , k ( pk ( x*))) around x*. Here, as in the discussion for the expectation opti mization model, it should be also stressed to the decision maker that any improv ement of one membership function can be achieved only at the expense of at least one of the other membership functions. 6. FRACTILE CRITERION OPTIMIZATION MODEL In this section, we discuss a fractile criterion model for the multi-objective c hance constrained programming problem (Sakawa et al., 2001), which aims to find the minimal value of multiple objective functions such that the probability of o btaining such a result is greater than or equal to some given thresholds under c hance constrained conditions.

Fuzzy Muti-objective Stochastic Linear Programming 397 Substituting the minimization of the objective functions zl ( x, ) , l 1,… ,k in E q. (4) for the minimization of values f l , l 1,… , k such that the probability of obtaining such result is greater than or equal to some given thresholds l under a chance constrained condition, the problem can be converted as Minimize f 1 Minimize f k subject to Pr c1 ( ) x Pr ck ( ) x ˆ Ax b x 0 f1 fk 1 (45) k where l (1/ 2, 1), l 1,… , k is assumed to guarantee the convexity of the finally reduced problem. In Eq. (45), we assume that cl ( ), l 1,… ,k are Gaussian random variable vectors. Then, the constraints Pr[ cl ( ) x fl ] l, l 1, … , k (46) are transformed as c ( ) x cl x Pr l xT Vl x f l cl x xT Vl x Pr[cl ( ) x fl ] l l. (47) Since random variables cl ( )x T cl x , l 1,… ,k (48) x Vl x in the above conditions, are standard Gaussian random variables with mean 0 and variance 12 , the conditions Eq. (46) are reduced to the following conditions:

398 M. Sakawa and K. Kato f l cl x xT Vl x l , l 1,…, k f l cl x xT Vl x K l , l 1,…, k fl cl x K l xT Vl x , l 1,…, k where ( ) is the distribution function of a standard Gaussian random 1 ( l ) . B ased on the above discussion, Eq. (45) variable and K l inf can be transformed i nto the following problem: Minimize f1 Minimize f k subject to c1x K 1 xT V1 x x T Vk x f1 (49) fk ck x K k Ax x 0 ˆ b . Equivalently, Eq. (49) can be rewritten as Minimize f1 (x ) minimize f k (x ) subject to A x x 0 ˆ b c1x K 1 ck x K k x T V1 x x T Vk x . (50)

Fuzzy Muti-objective Stochastic Linear Programming 399 Furthermore, each of the objective functions in Eq. (50) is convex since each of K l , l 1,…, k is positive from l ( 1 / 2 , 1 ) . Therefore, Eq. (50) is a multio bjective convex programming problem. In the following discussion, for notational convenience, the feasible region of Eq. (50) is denoted by X. In order to consi der the imprecise nature of the decision maker s judgments for each objective fu nction f l ( x) cl x K l xT Vl x in Eq. (50), if we introduce the fuzzy goals such as “ f l ( x ) should be substan tially less than or equal to a certain value”, (50) can be rewritten as: Maximize ( 1( f1( x )),… , k ( f k ( x ))) x X (51) where l ( ) is a membership function to quantify a fuzzy goal for the l th objec tive function in Eq. (50) and it is assumed to be concave. The shape of a typica l membership function is shown in Figure 4. Figure 4. An example of a membership function l( f l ( x )) For the decision maker’s reference membership levels l , l 1,… ,k , the correspondin g M-Pareto optimal solution, which is nearest to the requirements in the minimax sense or better than that if the reference membership levels are attainable, is obtained by solving the following minimax problem: M inimize max { l 1,… , k l l ( f l ( x ))} (52) subject to x X By introducing the auxiliary variable v, this problem can be equivalently transf ormed as

400 Minimize v subject to 1 1 ( f1 ( x)) ( f ( x)) k v M. Sakawa and K. Kato (53) k x X. k v If the optimal solution ( x*, v*) to Eq. (53) is not unique, the M-Pareto optima lity of x* is not guaranteed. In order to avoid the above situation, we consider the following augmented minimax problem Minimize v subject to 1 1 ( f1 ( x)) k i 1 k i 1 ( i i ( fi ( x))) v (54) k k ( f ( x)) k ( i i ( fi ( x))) v x X is a sufficiently small positive number. It should be noted that where the augme nted minimax problem (54) is a convex programming problem under the assumption t hat each of the membership functions 1,… ,k is nonincreasing and concave. l ( ), l Following the preceding discussions, we can now construct the interactive algor ithm in order to derive the satisfying solution for the decision maker from the M-Pareto optimal solution set. The steps marked with an asterisk involve interac tion with the decision maker. Interactive fuzzy satisfying method for fractile c riterion model Step 1. Ask the decision maker to specify the probability thresho lds ( 1 / 2 , 1 ) , l 1,… ,k and satisfying levels i , i 1, … , m for the chance l c onstrained condition in Eq. (1). Step 2. After calculating the individual minimu m f l ,min of f l ( x ) , l 1,… ,k in Eq. (50), ask the decision maker to determin e membership functions l ( f l ( x )) for objective functions in Eq. (50), which are nonincreasing and concave.

Fuzzy Muti-objective Stochastic Linear Programming 401 Step 3. Ask the decision maker to set the initial reference membership levels. I f it is difficult for the decision maker to specify them appropriately, set l 1, l 1, … , k . Step 4. Solve the augmented minimax problem Eq. (54) corresponding t o the reference membership levels l 1, l 1, … , k . Step 5. The decision maker is supplied with the corresponding MPareto optimal solution. If the decision maker is satisfied with the current membership function values of the M-Pareto optimal solution, stop. Otherwise, ask the decision maker to update the reference membe rship levels l 1, l 1,… ,k by considering the current membership function values l ( f l ( x*)) , and return to Step 4. Here it should be stressed to the decision maker that any improvement of one membership function can be achieved only at t he expense of at least one of the other membership functions. 7. NUMERICAL EXAMPLE In this section, being limited by space, we only present an illustrative numeric al example of an interactive fuzzy satisfying method using the 1 2 1 2 Minimize (c1 t1 ( )c1 )x ( 1 t1 ( ) 1 ) 2 2 Minimize (c1 t 2 ( )c2 )x ( 1 t 2 ( ) 2 ) 2 2 2 2 Minimize (c1 t 3 ( )c3 )x ( 1 t 3 ( ) 3 ) 3 3 subject to a 1x b1 ( ) a2 x b 2 ( ) a3 x b3 ( ) a4 x b 4 ( ) a5 x b5 ( ) a6 x a7 x x 0 b6 ( ) b7 ( ) (55)

402 M. Sakawa and K. Kato probability maximization model. Concerning numerical examples for other models, the interested readers might refer to the corresponding papers (Sakawa et al., 2 001;2000; 2002; 2004; Sakawa et al., 2003b; Sakawa and Kato, 2002). Consider the following multi-objective linear programming problem involving random variable coefficients (3 objectives, 10 variables, and 7 constraints). In this problem, t 1 ( ) , t 2 ( ) , and t 3 ( ) are Gaussian random variables N (4, 22 ) , N (3,3 2 ) , and N (3, 22 ) , where N ( , 2 ) stands for a Gaussian random variable hav ing mean and variance 2 . The right-hand side bi ( ) , i 1,… ,7 are also Gaussian random variables N (164,302 ) , N ( 190, 202 ) , N ( 184,152 ) , N (99, 222 ) , N (150,17 2 ) , N (154,352 ) and N (142, 422 ) . On the other hand, constant coe fficients in (55) are shown in Table 1 and Table 2. Table 1. Constant Coefficients of Objective Function in Eq. (55) c1 1 19 3 12 1 –18 2 48 2 –46 2 –26 1 21 2 –23 4 –22 3 10 1 –38 2 –28 2 18 4 –33 2 –15 1 35 3 –48 1 –29 2 46 1 12 2 –10 3 11 2 8 1 –19 3 24 4 19 2 –17 2 33 2 20 1 –28 1 1 1 2 1 1 2 2 2 –18 5 –27 6 –10 4 c1 2 c2 c2 1 2 1 c3 1 3 2 3

c3 2 First, according to Step 1, the decision maker determines the satisfying levels i , i 1,… ,7 for each of the constraints in Eq. (55). The hypothetical decision ma ker in this example specifies the satisfying levels as T ( 1 , 2 , 3 , 4 , 5 , 6 , 7 )T (0.85, 0.95, 0.80, 0.90, 0.85, 0.80, 0.90) . Second, according to Step 2 , the individual minimum zl ,min and maximum zl ,max of objective functions E zl ( x , ) , l 1,… ,k , are calculated under the chance constrained conditions corre sponding to the satisfying levels. Each value is obtained as z 1 ,min 1819.571 , z1,max 4221.883 , z 2,min 919.647 . 286.617 , z 2,max 1380.041 , z3,min 1087.249 , z 3,m ax

Fuzzy Muti-objective Stochastic Linear Programming Table 2. Constant Coefficients of Constraints in Eq. (55). 403 a1 a2 a3 a4 a5 a6 a7 12 2 3 11 4 5 3 2 5 16 6 7 3 4 4 3 4 5 6 14 6 7 16 8 9 5 3 9 13 6 8 1 13 9 6 1 12 2 8 6 7 18 6 12 12 4 2 4 11 6 4 12 6 5 4 9 11 7 4 9 14 5 4 8 10 3 6 6 9 7 Third, according to Step 3, the individual minimum pl ,min and maximum pl ,max o f pl ( x ), l 1,… ,k in the multi-objective probability maximization problem Eq. ( 27) are calculated as p1,min 0.002 , p1,max 0.880 , p3,min 0.002 , and p2,min 0. 328 , p2,max 0.783 , p3,max 0.664 . The decision maker subjectively determines m embership functions to quantify fuzzy goals for objective functions. Here, the f ollowing linear membership function is adopted: l ( pl (x)) pl (x) pl ,1 pl ,0 pl ,0 l( ), l In this chapter, parameters pl ,1 , pl ,0 in linear membership functions 1,… ,k ar e determined as p1,1 p 2 ,1 p 3,1 p1 ( x 1,max ) p 2 ( x 2 ,m ax ) p 3 ( x 3,max ) 0.880, p1,0 0 .783, p 2 ,0 0.664, p 3,0 m in{ p1 ( x l ,max )} l 2 ,3 0.502 m in{ p 2 ( x l ,m ax )} l 1,3 0.060 m in{ p 3 ( x l ,m ax )} l 1,2

0.446 by using Zimmermann’s method (Zimmermann, 1978). According to Step 4, the decision maker specifies the initial reference membership levels ( 1 , 2 , 3 ) as ( 1.00 , 1.00 , 1.00 ) . Next, according to Step 5, in order to find the optimal solut ion x* to the minimax problem (24) for ( 1 , 2 , 3 ) = (1.00, 1.00, 1.00) , afte r v* is calculated by solving the problem (28) using the bisection method and

404 M. Sakawa and K. Kato phase one of the two-phase simplex method, the linear programming problem (31) i s solved by the simplex method. The obtained solution is shown at the second col umn in Table 3. Table 3. Process of Interaction Interaction 1 2 3 1st 1.000 1.000 1.000 15.590 2.120 0.000 0.254 0.000 6.247 0.207 14.176 1.612 17 .932 0.5747 0.5732 0.5733 0.719 0.474 0.571 0.060 0.831 2nd 1.000 1.000 0.900 15.665 2.328 0.000 0.042 0.000 6.282 0.142 14.079 1.301 17 .733 0.6177 0.6172 0.5170 0.736 0.506 0.559 0.060 0.801 3rd 0.950 1.000 0.900 15.789 2.389 0.000 0.071 0.000 6.388 0.155 13.998 1.236 17 .694 0.5948 0.6436 0.5435 0.727 0.525 0.565 0.060 0.816 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 1 ( p1 ( x )) 2 ( p 2 ( x )) 3 ( p 3 ( x )) p1 ( x ) p2 ( x ) p3 ( x ) 1 / 2 / 3 1 According to Step 6, the hypothetical decision maker cannot be satisfied with th is solution, particular, he wants to improve 1 ( ) , 2 ( ) at the sacrifice of 3 ( ) . Thus, the decision maker updates the reference membership levels to (1.00 , 1.00, 0.90) and returns to Step 5. The result for the updated reference member ship levels is shown at the third column in Table 3. The decision maker is still discontented with the value of 2 ( p2 ( x )) . Since the sensitivity of 1 ( p1 ( x )) to 2 ( p2 ( x )) is higher than that of 3 ( p3 ( x )) from the trade-off information 0.060 and 1/ 2 / 3 0.801 , he updates the reference membership level s to 1

Fuzzy Muti-objective Stochastic Linear Programming 405 (0.95, 1.00, 0.90) to improve 2 ( p2 ( x )) at the expense of 1 ( p1 ( x )) . By repetition of such interaction with the decision maker, in this example, a sati sfying solution is obtained at the third interaction. 8. SOME EXTENSIONS So far, we have discussed interactive fuzzy satisfying methods for multiobjectiv e stochastic linear programming problems by making use of several stochastic mod els in chance constrained programming. As an alternative approach, the authors h ave proposed an interactive fuzzy satisfying method through a simple recourse mo del (Sakawa et al., 2001). Extensions of the proposed methods to more general ca ses such as multiobjective stochastic integer programming problems can be found in our papers (Kato et al., 2004a; 2004b; Perkgoz et al., 2003; 2004). For more extensions to two-level stochastic linear programming problems, the readers migh t refer to our papers (Kato et al., 2004c; Sakawa et al., 2003a; Wang et al., 20 04). 9. CONCLUSION In this chapter, we focused on multi-objective linear programming problems invol ving random variable coefficients. For transforming the original stochastic prog ramming into deterministic ones, several stochastic models such as an expectatio n-optimization model, a variance minimization model, a probability maximization model, and a fractile criterion optimization model for chance constrained condit ions are introduced. As a fusion of stochastic approaches and fuzzy ones, assumi ng that the decision maker has fuzzy goals for each of the objective functions i n the transformed problems, several interactive fuzzy satisfying methods for der iving a satisfying solution for the decision maker from the Pareto optimal solut ion set are presented. Through illustrative numerical examples, the feasibility of the proposed methods are demonstrated REFERENCES Bellman, R.E., and Zadeh, L.A., 1970, Decision making in a fuzzy environment, Ma nagement Science, 17: 141 164.

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AN INTERACTIVE ALGORITHM FOR DECOMPOSING: THE PARAMETRIC SPACE IN FUZZY MULTI-OB JECTIVE DYNAMIC PROGRAMMING PROBLEMS Mahmoud A. Abo-Sinna1, A.H. Amer2, and Hend H. EL Sayed3 Department of Basic Engineering Science, Faculty of Engineering, EL-Menoufia Uni versity, Shebin EL-kom, Tanta, AL-Gharbia, Egypt 2Department of Mathematics, Fac ulty of Science, Helwan University, Cairo, Egypt 3Department of Mathematics, Fac ulty of Science, Helwan University, Cairo, Egypt 1 Abstract: The aim of this chapter is to study the stability of multi-objective dynamic pro gramming (MODP) problems with fuzzy parameters in the objective functions and in the constraints. These fuzzy parameters are characterized by fuzzy numbers. For such problems, the concept and notion of the stability set of the first kind in parametric nonlinear programming problems are redefined and analyzed qualitativ ely under the concept of -Pareto optimality. An interactive fuzzy decision-makin g algorithm for the determination of any subset of the parametric space that has the same corresponding -Pareto optimal solution is proposed. A numerical exampl e is given to illustrate the method developed in the chapter. Fuzzy sets, Monte Carlo simulation, grey-related analysis, data mining Key words: 1. INTRODUCTION Most practical vector optimization problems contain measured or estimated values that are represented by the different coefficients of the objectives and constr aints. Such values may not be accurate enough to the errors in measuring, or est imating these values can lead to a false solution or a solution far from the exa ct solution of the considered problem. So, if C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 409

410 M.A. Abo-Sinna et al. after solving the problem an error is discovered or some factors are changed tha t affect these coefficients, the problem has to be solved again. Stability analy sis covers this difficulty. It tells us what coefficients affect the solution gr eatly if they are changed and what coefficients have negligible effects on the s olution. In this chapter, we study the stability of multiobjective dynamic progr amming (MODP) problems with fuzzy parameters in the objective functions and in t he constraints. These fuzzy parameters are characterized by fuzzy numbers. For s uch problems, concept and notion of the stability set of the first kind in param etric nonlinear programming problems are -Pareto redefined and analyzed qualitat ively under the concept of optimality. An interactive fuzzy decision making algo rithm for the determination of any subset of the parametric space which has the same -Pareto optimal solution is proposed. A numerical corresponding example is given to illustrate the method presented. 2. PROBLEM FORMULATION In this chapter, the fuzzy multiobjective dynamic programming (FMODP) problem is considered. Fuzzy vector-minimization problem (FVMP) involving fuzzy parameters in the objective functions and in the constraints (see Abo-Sinna, 1998, 1992, 2 004; Bellman, 1957; Bellman and Dreyfus, 1962; Carraway et al., 1990; Chankong, 1981; Cohon, 1978; Deng Feng and Chuntian, 2004; Esogbue, 1983; Henig, 1983; Hus sein and Abo Sinna, 1993; 1995; Larson and Casti, 1978; 1982; Mangasarian, 1969; Osman and El-Banna, 1993; Saad, 1995; Su and Hsu, 1991; Tauxe et al., 1979) are selected: FVMP: Minimize (1) Fq f q1 x1 , a1 ,..., f qN xN , aN , q 1 ,..., Q , Q 2 subject to Gm g m1 x1 , b1 ,..., g mN xN , bN xn X n , n 1 ,..., N , 0 , m 1 ,.. ., M , (2)

An Interactive Algorithm for Decomposing 411 where for each n 1,...., N , X is a subset of R k n ; xn is a k n vector, the ob jective functions Fq , q 1,...,Q and the constraint functions Gm , m 1,..., M are convex real valued functions of the class c 1 on R N and f qn , g mn , q 1,..., Q , m 1,..., M , n 1,..., N are real valued functions on ~ ~ ~ ~ ~ ~ X n , and a a11 , a22 ,..., aqn , b b ,b ,...b , q 1,..., Q , n 1,..., N 11 22 qn represent the vectors of fuzzy parameters in the objective functions ~ ~ f qn xn ,aqn and in the constraint functions g mn xn ,bqn , respectively. These fuzzy p arameters are assumed to be characterized as the fuzzy numbers introduced by Dub ois and Prade (1980). It is appropriate to recall here that a real fuzzy number ~ is a convex continuous fuzzy subset of the p real line whose membership functi on ~ p 0 is defined by (see Dubois p and Prade, 1980; Sakawa and Yano, 1990; Zim merman, 1985; 1987]): 1. A continuous mapping from real set R to the closed inte rval [0,1], , p1 , 2. ~ p 0 for all p p 3. strictly increasing on p1 , p2 , 4. ~ ( p ) 1 for all p p2 , p3 , p 5. strictly decreasing on p3 , p4 . 6. ~ ( p ) 0 for all p p4 , p A possible shape of fuzzy number ~ is illustrated in Figure 1. Now, we p ~ ~ assume that aqn and bqn in the FVMP ~ membership functions are a qn ( aqn ) and are fuzzy numbers whose ~ respectively, for b qn bqn ~ ~ simplicity are a a and b b . Here, we assume that the membership function ~ p is differentiable on p1 , p4 and the problem ( FVMP ) is p stable (see Rockafe llar, 1967; Sakawa and Yano, 1990). Now, we can introduce the definition of -lev el set or -cut of the fuzzy ~ ~ numbers a qn and bqn ( q 1 ,... ,Q , n 1 ,..., N ) (see Dubois and Prade, 1980 ). ~ p p 1 0 p1 p2 p3 p4 p Figure 1. Membership function of fuzzy number

412 M.A. Abo-Sinna et al. DEFINITION 1. ( - LEVEL SET) ~ The -level set of the fuzzy numbers aqn is define d as the ordinary set ~ for which the degree of all its component membership fun ctions L a 0, 1 exceeds the level L a a \ aqn aqn , q 1 ,..., Q , n 1 ,..., N Similarly, the -level set of the fuzzy numbers bqn is defined as the ~ ordinary set L b for which the degree of all its component membership 0 ,1 (see Zimmerman n, 1985;1987) functions exceeds the level L b b \ bqn ~ bqn , q 1 ,..., Q , n 1 ,..., N ~ Similarly, the -level set of the fuzzy numbers aqn and bqn is defined ~ ~ as t he ordinary set L a ,b for which the degree of all its component membership func tions exceeds the level 0, 1 ~ L a , b ( a , b) \ n 1,..., N aqn aqn , bqn bqn , q 1 ,..., Q , . Obviously, we have the following property for the level set: ~ ~ ~ ~ only if L 1 ( a ,b ) L 2 ( a ,b ) . 1 2 if and ~ As can be seen from the Definition 1, the -level set L a ,b is the set 0 , 1 .

of the closed intervals depending on the level For a certain degree of 0,1 , th e problem ( FVMP ) can be written in the following nonfuzzy parametric multiobje ctive dynamic programming problems (see Dauer and Osman, 1985; Osman and Dauer, ~ ~ 1983) depending on the parameters a ,b L a ,b , as was done by Sakawa and Ya no (1990): ( VMP ): ~ Minimize Fq f q1 x1 , a1 ,..., f qN xN , aN , q 1 ,..., Q, Q 2 (3)

An Interactive Algorithm for Decomposing subject to Gm g m 1 x1 ,b1 ,..., g mN x N ,bN xn X n , n 1 ,..., N , a ,b 0 , m 1 ..., M 413 (4) ~ ~ L a ,b Since (FVMP) is stable, the problem ( -VMP) is stable. It should be emphasized h ere that in the problem ( -VMP ), the parameters a and b are treated as decision variables rather than as constants. Separability and monotonicity of functions have been used for deriving the recursive formula of dynamic programming (see Ab o-Sinna and Hussein 1994; 1995). Definition of these properties for the problem ( VMP ) is given below. DEFINITION 2. (SEPARABILITY AND MONOTONICITY) The object ive function Fq is said to be separable if there exist n functions Fqn , n 1 ,.. ., N , defined on R n and functions Q q , n 2 ,..., N , 2 defined on R satisfyin g, for n 2 ,..., N , Fqn f q 1 x1 ,a1 ,..., f qn xn ,an n Qq Fqn 1 f q 1 x1 ,a1 ,..., f qn 1 xn 1 ,a n 1 , f qn xn ,a n (5) and FqN f q 1 x1 , a1 ,..., f qN x N , a N Fq f q 1 x1 , a1 ,..., f qN x N , a N . Similarly, the constraint function Gm is separable, if there exist n n functions Gm , n 1 ,..., N , defined on R n and functions m , n 2 ,..., N , 2 satisfying, for n 2 ,..., N defined on R n Gm g m1 x1 ,b1 ,..., g mn xn ,bn n m n Gm 1 g m 1 x1 ,b1 ,..., g mn 1 xn 1 ,bn 1 , g mn xn ,bn

and N Gm g m 1 x1 ,b1 ,..., g mN x N ,bN Gm g m 1 x1 ,b1 ,..., g mN x N ,bN

414 M.A. Abo-Sinna et al. If all objective and constraint functions are separable, we say that the n n pro blem - VMP is separable. Moreover, the functions q and q are called the separati ng functions of F and G . Furthermore, the separation of the problem - VMP is sa id to be n n monotone if all functions q and q are strictly increasing with resp ect to the first argument for each fixed second argument. Specifically, for each y R n q s,y s,y n q s , y iff s n m s s and n m s , y iff s For every q 1 ,..., Q m 1 ,..., M and n 2 ,..., N . -level set of the fuzzy numb ers (see Based on the definition of Kacprzyk and Orlovski, 1987; Orlovski, 1984; Zadeh, 1963) the concept of -Pareto optimal solution to the problem VMP is intr oduced in the following definition (see Sakawa and Yano, 1990). DEFINITION 3. ( -PARETO OPTIMAL SOLUTION) 0 A point x 0 x1 ,..., x 0 is said to be an -Pareto op timal solution to N VMP , if and only if there does not exist another the proble m x x1 ,..., x N , a ,b ~ ~ L ( a ,b ) such that 0 0 Fq f q 1 x1 , a1 ,..., f qN x 0 , a 0 N N Fq f q 1 x1 , a1 ,..., f qN x N , a N For all q and Fr f r 1 x1 , a1 ,..., f rN x N , a N 0 0 Fr f r 1 x1 , a1 ,..., f rN x 0 , a 0 N N For at least one index r 1 , 2 ,...,Q , where the corresponding values of ~ ~ pa rameters ( a 0 ,b0 ) L a ,b are called -level optimal parameters. VMP is separab le and the separation Assumption 1. The problem is monotone. ~ ~ is compact and Assumption 2. For every n , X n L a ,b n f ( x , a ),..., Fq q 1 1 1 f qn ( xn , a n ) , q 1 ,...,Q is continuous functions of n x1 ,..., xn , a1 ,...,an , and G m ( g m1 ( x1 ,b1 ) ... , g mn ( xn ,bn ) ), m 1 ,..., M is continuous functions of x1 ,..., xn and b1 ,...,bn .

An Interactive Algorithm for Decomposing 415 The problem VMP will be treated using one of the existing parametric approaches, i.e., by considering the following nonlinear program with scalar objective (see Chankong and Haimes, 1983) VMP : Minimize Q q Fq q 1 (6) f q 1 x1 , a1 ,..., f qN x N , a N subject to Gm g m 1 x1 ,b1 ,..., g mN x N ,bN 0 , m 1,..., M ( x1 X 1 ) ,..., x n X n , n 1 ,..., N , a ,b RQ \ Q q 1 q ~ ~ L a ,b , q for some 1, 0 . VMP implies the It is easy to see that the stability of the problem stability of the problem VMP . It is well known that x 0 is an Pareto optimal solution of th e problem VMP with the corresponding -level 0 optimal parameters 0 ( a 0 ,b 0 ) ~ ~ L a ,b if there exists VMP 0 0 such that x is the unique optimal solution of ~ ~ 0 0 if there exists 0 0 , provided every X n L a ,b is 1 ,..., Q 0 , 0 closed and convex . Let us suppose that every Fq is additive, i.e., for q 1,..., Q, (see Abo-Sinna, 1998) Fq f q 1 x1 , a1 ,..., f qN x N , a N f q 1 x1 , a1 ... f QN x N , a N .

Then the objective function in the problem N Q q n 1q 1 VMP becomes (7) f qn xn ,an = N n 1 f n xn , an

416 M.A. Abo-Sinna et al. If we define real valued functions Bn 0 and z z1 ,... z M by 1 ,..., Q n , z for each n 1,..., N , each Bn ,z min i 1 n fi xi , ai \ Gm g m1 x1 , b1 ,..., g mn xn , bn zm , m 1 ,..., M , x1 X 1 ,..., xn Xn , ( a, b) L a, b Now, we can obtain the recursive relations for n Bn ,z min x n X n ,( a ,b ) L ~ ~ a ,b 2,..., N : f n xn ,an Bn 1 ,z n 1 xn , z where z n 1 xn , z n z1 1 n xn , z ,..., z M 1 xn , z .

n n Assuming monotonicity of Gm , let z m 1 be defined by n zm 1 xn , z n R ; Gm sup , g mn xn , bn zm , bn ,..., bn L b m 1 ,..., M . THEOREM 1. 0 0 Suppose that Assumption 1 and Assumption 2 hold. Let x1 ,..., xn be any 0 0 -Pareto optimal solution of problem Bn , z for some , where the corresponding -l evel optimal parameters ~ 0 0 0 0 0 0 ~ ~ ~ a1 ,..., a n , b1 ,..., bn L a1 ,... , an , b1 ,..., bn . Then x1 ,..., xn 1 is an 0 n 1 Pareto optimal solution of p roblem Bn 1 , z xn , z , where the corresponding -level optimal parameters ~ ~ 0 0 0 0 ~ ~ a1 ,..., an 1 , b1 ,..., bn 1 L a1 ,..., an 1 , b1 ,..., bn 1 . The p roof of this theorem is much like that of Theorem 1 in (Mine and Fukushima, 1979 ). Using the recursive relations (2) for various values of we may find a VMP by set of -Pareto optimal solution of the problem obtaining Bn 0 ,0 .

An Interactive Algorithm for Decomposing 417 2.1 The Stability Set of the First Kind DEFINITION 4. with a corresponding -Pareto optimal Suppose that a certain 0 0 ~ ~ VMP , where a 0 ,b0 L a ,b are the solution x of the problem corresponding -le vel optimal parameters. Then the stability set of the 0 first kind of the proble m VMP corresponding to x , which is 0 0 0 denoted by S x , a , b , is defined by : S x 0 , a 0 ,b0 \ x 0 is an -Pareto optimal solution of the problem 0 0 ( VMP ) with the corresponding -level optimal parameters a ,b . THEOREM 2. If the func tions F and G are convex, and 0 0 0 ~ a a , ~ b b are concave functions, then the set S x , a ,b , which is the stability set of the first VMP corresponding to the kind of the problem -Pareto optimal solution x 0 with the -level optimal parameters a 0 ,b0 ~ ~ L a ,b , is * * * convex and S x 0 , a 0 ,b0 0 is closed. Furthermore, if S x ,a ,b is the V MP corresponding to the stability set of the first kind of the problem -Pareto optimal solution x* with the -level optimal parameters ~ ~ a* ,b* L a ,b and int S x 0 , a 0 ,b 0 S x* , a* ,b* , then S x 0 , a 0 ,b0 = S x* ,a* ,b* . The proof of this theorem is similar to the one in Osman (see Caplin and Kornblu th, 1957). REMARK 1. (Osman, 1977) It must be noted that the above properties of the stability set of the first kind still hold if the continuity and differenti ability assumptions that are imposed on F and G are relaxed. 2.2 Let 0 Determination of the Stability Set of the First Kind with an -Pareto optimal solution ( x 0 ) of the problem VMP with the correspondi ng -level optimal parameters ~ ~ a 0 ,b 0 L a ,b , then according to the Kuhn-Tu cker necessary optimality VMP , it follows conditions (see Mangarasian, 1969), f or the problem q q M that there exists , 0, U R , U 0 , V R , w R , V 0, w 0 , s uch that

418 T M.A. Abo-Sinna et al. G 0 0 F 0 0 (x , a ) U T (x ,b ) x x F 0 0 (x , a ) V T a G 0 0 x ,b b a a 0 (8) T a (a 0 ) 0 (9) UT WT b b b0 0 (10) G x 0 , b0 0, a0 0, b b0 0 , U T G x0 , b0 0 VT where sets: 0 a

a0 0, WT b b0 0 T stands for the transpose of the vector . Denote the following A x ,b J a 0 0 m \ Gm g m 1 x1 ,b1 ,..., g mN x N ,bN q ,n \ q ,n \ ~ a ~ b 0 0 0 0 0 , m 1 ,..., M aq 0 bq 0 and J b0 . Then we have the following three linear independent systems of equations: T f 0 0 x ,a x Q q q 1 m m A x0 , b0 Gm 0 0 x , b x aq 0 (11) Fq aq

x 0, a 0 Vq aq 0 aq 0 (12) UT J a 0 , Vq Gm 0 0 x ,b bq J a 0 ; wq Wq 0, q bq bq bq 0 J b0 (13) 0, q J b 0 , wq 0, q

An Interactive Algorithm for Decomposing 419 System (11) represents the first group of the Kuhn Tucker conditions and it can be rewritten in the following matrix form : C where matrix, D

0 matrix, ij (14) C cij R , Q is R K an s Q D d is s an h h k 0, 0 and 0, U R , V R , where 0 0 s, h are the cardinalities of A x ,b and J , r espectively . Suppose di j 0, j 1,..., K , i I 1, 2,..., s , where the cardinal number of I is assumed to be equal to s l . Then we ignore for moment these row s and consider the remaining system which will have the form C D 0 (15) Here C and D are matrices of order l Q and l k , respectively. Therefore system (11) together with the condition Q j 1 Ci

j

j 0, i I gives system (14), which is equivalent to system (11); hence we give the followi ng two propositions (see Zeleny , 1973; 1982): PROPOSITION 1. If K l , then T T C T ( D1 ) Q 1 j S x , a ,b 0 0 0 0 j \ j 1 ,...,l , j 1 C

0 ,i I (16) where D D1 D2 , D1 and D2 are respectively l l and l k l matrices and . j is the element in the jth column of the row vector PROPOSITION 2. If K l , then

ij

420 M.A. Abo-Sinna et al. \ S x 0 , a 0 , b0 T T C2 C1T ( D1T ) T 1 T D2 1 j j 0 (17) j 1 ,..., k l , Q C1T D1T j j 0 i I j 1, ..., k , j 1 Ci 0, REMARK 2. If is normalized by the condition Q q 1 q 1, then we can add this condition to the set S x 0 , a 0 ,b0 in any one of its form. 2.3 An Algorithm Now, we can construct an algorithm to determine the stability set VMP ) as follo ws. S x 0 , a 0 , b 0 of the problem ( Step 1. Elicit a membership function from the decision maker for each ~ ~ of the fuzzy numbers a and b in the problem FVM P . Step 2. Ask the decision maker to select the initial values of (0 1 ). Step 3. Construct the parametric multi-objective dynamic program VMP . general, as the vector minimization problem Step 4. Ask the decision maker to choose cert ain 0 and by using the recursive relations (2), the decision maker approach can be used to VMP by obtain an -Pareto optimal solution x 0 of the problem ~ obtaining Bn , 0 , n 1,..., N . Suppose that a 0 ,b0 L a ,b is the correspondi ng -level optimal parameters (using any available nonlinear programming package,

for example, GINO at each stage). Step 5. Substitute with x 0 , a 0 , b 0 in th e Kuhn Tucker necessary conditions, we obtain system (11), and system (15) can b e easily found. Also system (12) can be solved by Gauss-elimination. Step 6. Acc ording to the values of the Lagrange multipliers, we get a) if s m k l , then S x 0 , a 0 ,b 0 t 0 \ t 0 ; b) if k l , then S x 0 , a 0 , b 0 is ~ given by (16), andc) if k l , then S x 0 , a 0 , b 0 is given by Eq. (17)

An Interactive Algorithm for Decomposing 421 Step 7. If the DM is satisfied with current solutions, stop. Otherwise, ask the decision maker to update the degree ( ) 0 ,1 and return to Step 3. 2.4 Numerical Example Let us consider the following multi-objective dynamic programming problem with f uzzy parameters in the objective functions (in fact, this problem has three stag es and three objectives), namely, the fuzzy vector minimization problem is writt en as follows ( FVMP ): ~ Minimize f 1 x , a1 ~ Minimize f x , a 2 ~ x1 a11 2 2 x2 2 x3 2 x1 1 2x1 2 x2 a22 2 x3 2 2 ~ Minimize f 3 x , a3 2 x2 x3 a33 . subject to M1 x R 3 \ x1 x2 x3 3, x j 0, j 1, 2, 3 Let the fuzzy parameters be characterized by the following fuzzy numbers: a11 0,1,3,5 , a22 0,1, 4, 6 ,

a33 3,5,9,10 . ~ Assume that membership function for each fuzzy number a in problem (FVMP) is d efined by 0, 1 ~ a(a) 1 0, 1, a p3 p4 p3 2 a a p2 p1 p2 2 p1 p2 p3. p4 , p1 p2 a a a a , p3 p4 Consider the given by – level sets or – cuts of the fuzzy numbers, which are

422 ~ a 11 ~ a 22 ~ a 33 M.A. Abo-Sinna et al. a11 a22 a33 0.36 ; then we get 0.2 0.36 ; then we get 0.4 0.36 ; then we get 3.4 a11 a22 a11 4.6 5.6 9.8 (See Figure 1). a ~ a 1 0 p1 p2 p3 p4 a Figure 2. – cuts of the fuzzy numbers. ~ a11 a11 1 0.36 0 0.2 1 3 p3 p1 p2 4.6 5 p4 a11

Figure 3. – cuts of the fuzzy numbers a 22 a 22 1 0.36 0 r1 0.4 1 r2 4 r3 5.6 6 r4 a 22 Figure 4. – cuts of the fuzzy numbers

An Interactive Algorithm for Decomposing a33 423 a 33 1 0.36 0 3 h1 5 h2 9 h3 10 h4 a 33 Figure 5. – cuts of the fuzzy numbers The nonfuzzy written as follows: ( VMP ) : - vector minimization problem ( VMP ) can be Minimize f 1 ( x , a1 ), f 2 ( x , a 2 ), f 3 ( x , a3 ) subject to x1 3.4 where f 1 x , a1 f 2 x ,a2 f 3 x ,a3 x1 a11 2 2 x2 a33 x3 3 , 9.8 , 0.2 a11 4.6 , 0.4 0 a22

5.6 (18) x1 , x2 , x3 x2 x2 x3 2 x3 a 22 a33 2 x1 1 2 x1 2 x3 2 2 x2 2 2 VMP ) satisfies Assumptions 1 and 2. It is easy to see that ( Therefore a dynami c programming approach can be applied for VMP . characterizing the -Pareto optim al solution of the problem Using the weighting method (Chankong and Hamines, 198 3) then the VMP becomes problem ( VMP ) : 3 q q 1 Minimize f q x ,a . Subject to the set of constraints Eq. (18)

424 M.A. Abo-Sinna et al. i) At 0 1/ 3, 1/ 3,1/ 3, , the dynamic programming approach has the following st eps: Step 1. B1 0 ,0 Minimize 1/ 3 x1 a11 3 q 1 0 q f q 1 x1 , a11 / x1 2 3 , 0.2 a11 4.6 , x1 0 x1 0 Minimize 2 1/ 3 x1 1 2/ 3x1 x1 3, 0.2 a11 4.6, by using GINO package, the x ,a 0 1 0 11 -Pareto optimal solution to B1 0 ,0 be 0.1 ,0.2 . Step 2. 3 B2 0 , 0 Min

q 1 0 q q1 f 0 x10 , a11 f q 2 x2 , a22 \ x10 2 1 3 2 x2 x2 3 , x10 , x2 0, 0.4 a22 5.6 Min 0.34 1 3 2 x2 1 3 x2 a22 \ 0.1 x2 3, 0.4 0 a22 5.6, x2 0 Hence the -Pareto optimal solution to B1 0.1, 0.0 , 0.2 ,0.4 . ,0 0 0 0 0 is x1 , x2 , a11 , a22 Step 3. 3 B3 0 , 0 =Minimum q 1

0 q q1 f 0 x10 , a11 0 0 f q 2 x2 , a22 f q 3 x3 , a33 \ x10 0 x2 x3 3 3.4 a33 0 0 9.8 , x1 , x2 , x3 0 . Thus B3 0 , 0 Minimum 0.39333 1 3 2 x3 1 3 x3 2 2 1 3 x3 a33 2 \ 0.1 0.0 x3 3, 3.4 a33 9.8 , x3

0 0 0 0 0 0 0 and x1 , x2 , x3 , a11 , a22 , a33 0.1,0.0 ,1.8 ,0.2 ,0.4 , 3.4 is t he -Pareto optimal solution to ( VMP ) and the optimum objective value equals 2. 34. ii) Determining the ce all functions Gm fuzzy number b , it tability set of the stability set of the first kind to the problem ( VMP ) : Sin ( m 1,2 ,3 ,4 ) in the problem ( FVMP ) do not ~ appear the is easily seen that the set S x 0 , a 0 ,b0 , which is the s first kind of the problem ( VMP )

An Interactive Algorithm for Decomposing 425 corresponding to the -Pareto optimal solution x 0 ( 0.1,0 ,0 ) with the Level op timal parameters a 0 ( 0.2 ,0.4 ,3.4 ) is S x 0 ,a 0 . Therefore, in what follow s, we will determine the set S x 0 ,a 0 . From the Kuhn Tucker necessary optimal ity conditions (system (19) and system (12)) we get: 0.2 1 1.8 0.8 2 2 2 3 1 2 0 0 (19) 3.6 0 .2 1 1 0.4 0.8 2 2 3.2 3 3 3.2 3 v2 v3 v4 0 0 0 0 (20) System (19) will be the same as system (21) and (22), i.e., s l k 3 ; thus 0.2 0 3.6 T T ( D1 C 1.8 0.8 0.4 0.2 2 0 ,D 3.2 0 3.6 0.4 3.2

D1 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 0.2 0 3.6 0.4 3.2 1.8 0.8 2 0 3.6 0.4 3.2 C ) 1 1.8 0.8 2 0 C (D ) 1 1, 2, 3 0.2 0 1.8 0.8 2 0 2, 0.2 1 1.8 2 2 3, 0.8 8 3.6 , 1 0.4 2 3.2 3 . We get 1 9 2 10 3 , system (20) as follows: v1 0 , 0.2 1 2 3

9 1 1 2 3 1 . Now we can solve v2 0 , 0.8 2 v3 0 , 3.2 3 v4 0.

426 M.A. Abo-Sinna et al. From 1 2 3 1 3 5 v2 10 3 v3 10 32 v4 we have v2 1 15 , v3 8 30 4 15 and v4 32 30 16 15 . If c1 0 a1 c2 , d 1 0 a2 d 2 ,e1 0 a3 e2 , then V1 0.2 c2 V2 c1 0.2 V3 0.4 d 2 V4 d1 0.4 V5 3.4 e2 v6 (e1 3.4) 0 , i.e. 0.2 c2 , c1 0.2, 0.4 d2 , d2 0.4, 3.4 e2 , e2

3.4. The x ,a 0 0 stability set of the first kind corresponding to 0.1,0.0 ,1.8 ,0.2 ,0.4 , 3.4 takes the form: S 0.1, 0.0, 1.8, 0.2, 0.4, 3.4 , p, r , h \ p1 r1 h1 p2 1 4 p1 r2 1 15r1 h2 17 4 h1 r3 h3 1 9 p3 2 10 3 , p1 2 8 0.2, 3 9 1, 1 2 3 1 p4 , 0 r4 , 0 r1 0.4 h4 , 0 h 3.4 It must be observed here that, if the decision maker is not satisfied with of th e Pareto optimal solution, it is the current value of the degree possible for th e decision maker to continue the same procedure in this manner until the decisio

n maker is satisfied with the current value of the Pareto optimal solution. degr ee of the 3. CONCLUSIONS In this chapter, the stability of multi-objective dynamic programming (MODP) pro blems with fuzzy parameters in the objective functions and in the constraints ha s been studied. An interactive fuzzy decision-making algorithm for the determina tion of any subset of the parametric space that Pareto optimal solution has been has the same corresponding proposed. Interactive algorithms have a significant potential for the fuzzy research in the future.

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GOAL PROGRAMMING APPROACHES FOR SOLVING FUZZY INTEGER MULTI-CRITERIA DECISION-MA KING PROBLEMS Omar M. Saad Department of Mathematics, College of Science, Qatar University, Doha, Qatar Abstract: Multicriteria decision making can be divided into two parts: multi-attribute dec ision analysis and multi-criteria optimization. When the number of the feasible alternatives is large, we use multi-criteria optimization. On the other hand, mu lti-attribute decision analysis is most often applicable to problems with a smal l number of alternatives in an environment of uncertainty. In this chapter, a go al programming approach was analyzed to solve fuzzy integer multi-criteria decis ion-making problems. Goal programming, integer multi-criteria decision-making pr oblem, iterative goal programming approach, fuzzy integer multi-criteria decisio n making problem Key words: 1. INTRODUCTION The term “multi-criteria decision making” (MCDM) encompasses a wide variety of probl ems. Multi-criteria decision making is concerned with the methods and procedures by which multi-criteria can be formally incorporated into the analytical proces s. Multi-criteria decision making has, however, two distinct halves: one half, i s multi-attribute decision analysis, and the other is multi-criteria optimizatio n (multi-objective mathematical programming). Multi-attribute decision analysis is most often applicable to problems with a small number of alternatives in an e nvironment of uncertainty. Multi-criteria optimization is often applied to deter ministic problems in which the number of feasible alternatives is large. C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 431

432 O.M. Saad In recent years research has been carried out in solving multi-criteria integer programming problems, but whereas some has been classified as such, some has app eared in terms such as decision theory. Treating integer multi-criteria decision -making problems can be classified into three main approaches: vector optimizati on (multi-objective optimization), goal programming, and interactive approaches. Most of the current research is directed mainly toward the interacting approach es trying to avoid the drawbacks in the other two approaches. Also, the current research includes the stochastic and fuzzy cases. Most decision problems have mu ltiple objectives that cannot be optimized simultaneously due to the inherent co nflict between these objectives. Such problems involve making trade-off decision s to get the “best compromise” solution. Goal programming is a powerful approach tha t has been proposed for the modeling, solution, and analysis of the multi-criter ia decision-making problems. There are a wide variety of goal programming models , including weighted goal programming (Charnes and Cooper, 1961; Ignizio, 1983) lexicographic goal programming, i.e., the use of the so-called “preemptive priorit y” concept (Ignizio, 1976; 1983), minimax goal programming includes fuzzy programm ing (Zimmerman, 1978) and interactive goal programming that is used to generate a subset of the nondominated solutions (Ignizio, 1981; Steuer, 1978). Since goal programming now encompasses any linear, integer, zero-one, or nonlinear multi-o bjective problem (for which preemptive priorities may be established), the field of applications is wide open. The recent increase in interest in this area has already led to a large number of and a wide variety of actual and proposed appli cations. For purpose of illustration, we list just a few of these below, and the reader is referred to (Ignizio, 1978): Aggregate planning and work force (Dauer and Osman, 1981). Qualitative programming for selecting decisions (Zahedi, 1987 ). Curve and response surface fitting (Ignizio, 1977). Media planning (Chranes e t al., 1968). Manpower planning (Chanes and Nilhaus, 1968). Program selection (S atterfield and Ignizio, 1974). Hospital administration (Lee, 1971). Academic res ource allocation (Schroeder, 1974). Municipal economic planning (Lee and Sevebec k, 1971). Transportation problems (Lee and Moore, 1973). Energy/water resources (Elchak and Raphael, 1977). Radar system design (Ignizio and Satterfield, 1977). Sonar system design (Wilson and Ignizio, 1977).

Goal Programming For Fuzzy Integer MCDM 433 Planning in wood products (Inoue and Eslick, 1975). Portfolio selection (Kumar a nd Philippatos, 1975) Determination of time standards (Mashimo, 1977). Developme nt of cost estimating relationship (Ignizio, Inpress). Urban renewal planning (L ee and Keown, 1976). Merger strategy (Salkin and Jones, 1972). Multi-plant/produ ct aggregate production loading (Johnson, 1976). BMD systems design (Ignizio and Satterfield, 1977). Multi-objective facility location (Harnett and Ignizio, 197 2). Free flight rockets (Ignizio, 1975a). Solar heating and cooling (Ignizio, 19 75b). Natural gas well sitting (Gochnour, 1976). Maintenance level determination (Younis, 1977). A Pennsylvania coal model (Kirtland et al., 1977). All of these applications have one thing in common: they could be forced onto a traditional single-objective model if one so wished. However, those investigating these prob lems believed that they truly involved multiple, conflicting objectives and were thus most naturally modeled as a goal programming problem (Ignizio, 1978). 2. INTEGER MULTICRITERIA DECISIONMAKING PROBLEM AS A GOAL PROGRAMMING MODEL The integer multi-criteria decision-making problem (IMCDM) can be formulated mat hematically as follows: (IMCDM): Maximize subject to Z(x) x X where Z: R n R k , Z(x) ( z1 (x), z 2 (x), ..., z k (x) ) is a vector-valued 1, 2 , ..., k which are real-valued objective criterion with zi (x), i functions an d X is feasible set. This set might be, for example, of the form: X { x R n Ax b, x 0 and integer }

434 O.M. Saad where A is an ( m n ) matrix of constraint function coefficients; x is an ( n 1) vector of the integer decision variables; b is an ( m 1) vector of constraint r ight-hand sides, whose components specify the available resource; and R n is the set of all ordered n-tuples of real numbers. It is assumed in problem (IMCDM) t hat the feasible set X is bounded. The imperative “maximize” in problem (IMCDM) is u nderstood to mean: Find the set of all solutions that have (roughly) the propert y that increasing the value of one objective z k (x) decreases the value of at l east one other objective function. This set is usually called an efficient (or n ondominated, noninferior, Pareto-optimal, functional-efficient) set. This set is a surrogate for an optimal solution to a usual optimization problem with a sing le objective function. The meaning of an efficient solution is given in the foll owing definition. DEFINITION 1. A point x* X is said to be an efficient solution of problem (IMCDM) if there exists no other x X such that Z(x) Z(x * ) and Z(x) Z(x * ) (see Chankong and Haims, 1983; Cohon, 1978; Geoffrion, 1968, Hwang and Masud, 1979). Now, let us express the ith objective function in the form: z i(x) cit x, where the superscript t denotes the transpose and ci is an n-vector defi ned as the vector of the coefficients of the ith objective function. In goal pro gramming, rather than attempting to optimize the objective criteria directly, th e decision maker sets to minimize the deviations between goals and levels of ach ievement within the given set of constraints. Thus, the objective function becom es the minimization of these deviations on the relative importance assigned to t hem. Problem (IMCDM) can be transformed into the following integer linear goal p rogramming model (ILGP) consisting of k goals: (ILGP): Find x to achieve: z1(x) h1 z 2(x) h2 . . . z k (x) hk subject to x X

Goal Programming For Fuzzy Integer MCDM 435 where h1 , h2 ,..., hk are scalars and represent the desired achievement levels of the objective functions that the decision maker wishes to attain provided tha t z*i hi z*i , i 1,2,...,k. * and z provide upper and lower bounds on the object ive Note that z * function values and hence are a great source of information fo r the decision maker. These bounds can easily be determined by solving: Maximize subject to zi (x) (1) Ax b, 0 and integer. x The solution of problem (1), (x* , z* ) , is known in the literature as the idea l i i solution. Let z ji z i(x j ); then z*i min z ji , {i} j 1,2,..., k . (2) DEFINITION 2. The goals are ranked as follows: if i j then goal i, cit x hi , ha s a higher priority than goal j , c tj x h j , ( see preemptive priorities Charn es and Cooper, 1961; Lee, 1972). 3. AN ITERATIVE GOAL PROGRAMMING APPROACH FOR SOLVING (IMCDM) PROBLEM In order to solve the integer linear goal program (ILGP) by the iteration algori thm developed in Dauer and Krueger (1977) together with the Gomory’s fractional cu t shown in Klein and Holm (1978, 1979), we first solve the integer linear optimi zation problem associated with the first goal viz:

436 O.M. Saad P1: minimize subject to t c1 x L1 d1 d1 b, 0, d1 d1 d1 0,x h1 , 0 , and integer Ax d1 and the overattainment, where d 1 and d 1 are the underattainment respectively, of the first goal where d 1 d 1 0. Suppose this problem has integer optimal valu e L* d 1 * d 1 * with at 1 least one value d 1 * or d 1 * nonzero. Now, the atta inment problem for goal 2 is equivalent to the integer optimization problem P2 , where P2 : minimize subject to t c2 x L2 d2 d2 d1 d1 b, 0, di d2 d2 d1 L , 0, x 0 , and integer, i 1,2. * 1 h2 , h1 , c x d1 Ax di t 1 Letting L* d 2 * d 2 * to denote the integer optimal value of problem P2 , 2 we can proceed to goal 3. The general attainment problem Pj for goal j is written a s

Goal Programming For Fuzzy Integer MCDM 437 Pj : minimize subject to cit x di Ax di di di b, 0, di 0,x 0, and integer, 1 i j di L, * i Lj dj dj hi , 1 1 i i j j 1 where d i and d i are the underattainment and the overattainment, respectively, of the ith goal level and d i d i 0. The integer optimal objective value of prob lem Pj , L*j , is the maximum degree of attainment for goal j subject to the max imum attainment of goals 1, 2,…, j 1. Notice that L*j 0 if and only if goal j is a ttained. Let x* be the optimal integer solution of the integer attainment proble m Pk associated with the minimum L*k ; then the solution of the ILGP is given by x* . The procedure used to solve the ILGP can be summarized as follows. 4. SEQUENTIAL INTEGER GOAL ATTAINMENT ALGORITHM Step 1. Formulate the ILGP corresponding to the (IMCDM) problem. Step 2. Solve t he integer attainment problem P1 for goal 1 using Gomory’s cutting-plane technique (Klein and Holm, 1978; 1979) and obtain L* . 1 Step 3. Set i = 2. Step 4. Using L* , L* ,..., L* 1 , solve the integer attainment problems Pi 1 2 i using the s ame cutting-plane technique used in step 2. Let L* denote the minimum. i Step 5. If i k , set i i 1 and go to step 4. Otherwise, go to step 6. * Step 6. Let x* ( x1 , x* ,..., x* ) denote the integer solution(s) of the 2 n attainment proble m Pk associated with the minimum L* . k The optimal integer solution(s) of the I LGP is then given by x* .

438 O.M. Saad 5. AN ILLUSTRATIVE EXAMPLE (CRISP CASE) In this section, we consider the following integer multi-criteria decisionmaking problem with two objective functions: (IMCDM) : Maximize subject to x X Z(x) ( z 1(x), z 2 (x) ) where X x R2 x1 x2 5, x1 x2 0 , 6 x1 2 x2 21, x1 , x2 0 and integer and z1 (x) z2 (x) 2 x1 x1 x2 2 x2 . Suppose that the decision maker specifies the first priority goal to be z 1(x) and the second priority goal to be z 2(x). Consequently, an equivalent integer linear goal program corresponding to the IMCDM problem can be written as follows: (ILGP): Goal 1: Goal 2: subject x X to Achieve Achieve 2 x1 x1 x2 2 x2 h1 h2 It is easy to see that the aspiration levels of the objectives h1(x) 7 and h2(x) , respectively. The integer linear attainment problem associated with the first goal is written as z 1(x) and z 2(x) are

Goal Programming For Fuzzy Integer MCDM P1: minimize subject to 2 x1 x1 x1 6 x1 x2 x2 x2 2 x2 5 0 21 0 and integer d1 d1 7 L1 d1 d1 439 d1 , d1 , x1 , x1 This problem can be solved using the following Gomory cuts, (see Klei and Holm, 1978;1979): 2 x1 x1 x2 x2 x1 7 4 3 and the maximum degree of attainment of problem P1 is L*1 0, with an optimal int eger solution x 1 ( 3,1) where d 1 0 and d 2 0. The attainment problem for goal 2 is equivalent to the integer optimization problem P2 , where P2 : minimize subject to x1 2 x1 d1 x1 x1 6 x1 i 1, 2 d1 x2 x2 2 x2 5 0 21 0 and integer, 2 x2 x2 0 d2 d1 d2 d1 6 7 L2 d2 d2 d i , d i , x1 , x2 The initial solution x 1 ( 3, 1 ) , d 1 x1 2 x2 5. 0 and d 2 0 yields a goal 2 value

440 O.M. Saad The maximum degree of attainment of goal 2 is L*2 1 with an 2 optimal integer so lution x ( 3, 1 ), where d 2 1 and d 2 0. Therefore, the optimal integer solutio n of the ILGP is given by x* L L * 1 * 2 ( 3, 1) d1 d2 0 , d1 1, d2 0 0 0, 1, with with 6. FUZZY INTEGER MULTI-CRITERIA DECISION-MAKING PROBEM (FIMCDM) In this section, we begin by introducing the following fuzzy integer multicriter ia decision-making problem with fuzzy parameters in the right-hand side of the c onstraints as ( FIMCDM ) : Maximize Z(x) subject to x X( ) where X( ) x R n / g r ( x) r , (r 1, 2, ...., m), x 0 and integer and Z : R n R k , Z(x) = (z1(x), z2(x),…, zk(x)) is a vector-valued criterion with zi(x), (i=1,2,..,k) are real-valued linear objective functions, ( 1, 2 ,..., m ) t is a vector of fuzzy parameters, and Rn is the set of all ordered n-tuples of real numbers. Furthermore, the constraints functions gr(x), (r =1, 2,…, m) are assumed to be linear. Now, going back to problem ( FIMCDM ) , w e can write an associated fuzzy integer linear goal programming model (FILGP) co nsisting of k goals and having R m a vector of fuzzy parameters in the right-han d side of the constraints. This model may be expressed as

Goal Programming For Fuzzy Integer MCDM (FILGP) 441 : z1(x) = h1, z2(x) = h2 Achieve: zk(x) = hk and the constraints are given by gr(x) x r , (r = 1, 2, …, m) 0 and integer where h1, h2,…,hk are scalars and represent the aspiration levels associated with the objectives z1(x), z2(x),…, zk(x), respectively. 7. FUZZY CONCEPTS The fuzzy theory has been advanced by L.A. Zadeh at the University of California in 1965. This theory proposes a mathematical technique for dealing with impreci se concepts and problems that have many possible solutions. The concept of fuzzy mathematical programming on a general level was first proposed by Tanaka et al. (1974) in the framework of the fuzzy decision of Zadeh and Bellman (Zadeh, 1970 ). For the development that follows, we introduce some defintions concerning tra peziodal fuzzy numbers and their membership functions, which come from (Dubois a nd Prade, 1980) ,and that will be used throughout this part. It should be noted that an equivalent approach can be used in the triangular fuzzy numbers case. DE FINITION 3. A real fuzzy number a is a fuzzy subset from the real line R with me mbership function (a) that satisfies the following assumptions: a 1. 2. 3. ~ a ~ a ~ a a is a continuous mapping from R to the closed interval [0, 1], a =0 a ( , a1 ], a is strictly increasing and continuous on [a1, a2],

442 O.M. Saad ~ a ~ a ~ a 4. 5. 6. a =1 a [a2, a3], a is strictly decreasing and continuous on [ a3, a4], a =0 a [ a4, + ). where a1, a2, a3, a4 are real numbers and the fuzzy number a is denoted by ~ a = [a1, a2, a3, a4]. DEFINITION 4. ~ The fuzzy number a is a trapezoidal number, d enoted by [a1, a2, a3, ~ a4], and its membership function a a is given by (see F igure 1). 0 , 1 a a a a1 a2 a2 2 a1 a a a a2 a3 a4 , a1 a2 a 1 , 1 0 , a a4 a3 a3 2 , a3 otherwise. a (a) 1 0 a1 a2 a3 a4 a ~ Figure 1. Membership function of a fuzzy number a DEFINITION 5. ~ The -level set of the fuzzy number a is defined as the ordinary set ~ ) for which the degree of their membership function exceeds the L (a level [0, 1]:

Goal Programming For Fuzzy Integer MCDM 443 ~ L (a ) a R a (a) . For a certain degree * [0, 1] with the corresponding -level set of the ~ ~ fuzzy numbers v r , problem (FILGP )v can be understood as the following nonfuzzy int eger linear goal programming model written as: ( FILGP) : z1(x) = h1 z2(x) = h2 Achieve: zk(x) = hk subject to gr(x) r r , (r = 1, 2, …, m) = 1, 2,…, m) L ( ~r ) , (r v x 0 and integer ~ where L ( ~r ) is the - level set of the fuzzy parameters v r , (r = 1, 2, v …, m). We now rewrite problem ( FILGP) above in the following equivalent form: ( IL GP ) : z1(x) = h1 z2(x) = h2 Achieve: zk(x) = hk subject to

444 O.M. Saad gr(x) ( nr 0 ) r r , (r = 1, 2, …, m) ( Nr0 ) , (r = 1, 2,…, m) x 0 and integer ~ It should be noted that the constraint v r L ( v r ) , (r = 1, 2,…, m), has (0 (0 ) been replaced by the equivalent constraint n r N r , (r = 1, 2,…, m), r (0 (0 ) where n r and N r are lower and upper bounds on r . Taking into account re strictions gr(x) r , (r = 1, 2, …, m) and for the purpose of solving the integer inear goal program ( ILGP ) at * [0, 1], we N r(0) , (r = 1,2,…,m) for a certain egree = r r use the iterative approach developed in Dauer and Rrueger (1977) tog ether with the Gromory cuts shown in Klein and Holm (1978, 1979). First, we solv e the following integer linear optimization problem associated with the first go al, viz: P( r ) : 1 Minimize L1 = d 1 + d 1 subject to z1(x) + d 1 gr(x) r d 1 = h1 , (r = 1, 2, …, m) 0 and integer d 1 0 , d1 0 , x where d 1 and d 1 are the underattainment and the overattainment, respectively, of the first goal where d 1 d 1 = 0. Suppose this problem has integer optimal va lue L*1 = d 1 + d 1 with at least one value d 1 or d 1 nonzero. Now, the attainm ent problem for goal 2 is equivalent to the integer optimization P2 ( r ) , wher e P2 ( r ) : Minimize subject to L2= d 2 + d 2 z2(x) + d 2 d 2 = h2 ) ) l d

Goal Programming For Fuzzy Integer MCDM 445 z1(x) + d 1 d 1 = h1 d 1 + d 1 = L1 gr(x) r * , (r = 1, 2, …, m) 0 and integer, (i = 1,2) r d i 0 , di 0 , x Letting L*2 = d 2 + d 2 denotes the integer optimal value of P2 ( can proceed to goal 3. ) , we The general attainment problem P j ( r ) for goal j is written as Pj ( r ): Lj = d j + d j Minimize subject to zi(x) + d i d i = hi, (1 i j-1) j) * d i + d i = Li , (1 i gr(x) r , (r = 1, 2, …, m) 0 and integer, (1 i j) d i 0 , di 0 , x where d i and d i are the underattainment and the overattainment, respectively, of the ith goal level and d i d i = 0. The integer objective value of Pj ( r ) , L*j , is the maximum degree of attainment for goal j subject to the maximum att ainment of goals 1, 2,…, j 1. Notice that L*j = 0 if and only if goal j is attaine d. Let x* be the optimal integer solution of the integer attainment problem P j ( r ) associated with the minimum L*j , then the solution of the integer goal pr ogram ( ILGP ) is given by x* with = * [0, 1].

446 O.M. Saad 8. AN ITERATIVE GOAL PROGRAMMING APPROACH FOR SOLVING FIMCDM Now, we develop a solution algorithm to solve the fuzzy integer linear goal prog ram (FILGP) .The outline of this algorithm is as follows (Alg-II): Step 0. Set = *= 0. Step 1. Determine the points (a1, a2, a3, a4) for each fuzzy ~ parameter r , (r = 1, 2, …, m) in program (FILGP )v with the * ~ v for the vector of fuzzy c orresponding membership function a ~ ( 1 , 2 ,..., m ) t . parameters Step 2. Co nvert program (FILGP )~ into the nonfuzzy integer linear v goal program ( ILGP ) . N (0) , (r = 1, 2,…, m) and solve Step 3. Choose r r r problem P1 ( r ) using G omory’s cutting- plane method (Klein and Holm, 1978, 1979) and obtain L * . 1 Step 4. Set j =2. Step 5. Using L * , L * ,…, L *j 1 , solve P j ( r ) using the same 1 2 Gomory’s cutting-plane method used in step 3. Let L *j denotes the minimum. St ep 6. If j k, set j = j +1 and go to step 5. Otherwise, go to Step 7. Step 7. Le t x* denotes the optimal integer solution of problem Pj ( r ) associated with th e minimum L*j . Step 8. Set = ( *+ step) [0, 1], and go to Step 1. Step 9. Repea t again the above procedure until the interval [0, 1] is fully exhausted. Then s top. 9. AN ILLUSTRATIVE EXAMPLE (FUZZY CASE) Consider the following integer linear goal program involving fuzzy v v ~ paramet ers ( ~1 , ~2 , v 3 ) in the right-hand side of the constraints: (FILGP) : goal 1: goal 2: subject to Achieve Achieve 2x1 + x2 = h1 x1 + 2x2 = h2 x1 + x2 ~1 v ~ x1 + x2 v 2

Goal Programming For Fuzzy Integer MCDM ~ 6x1 + 2x2 v3 x1 0, x2 0 and integers. 447 Assume that the membership function corresponding to the fuzzy parameters is in the form: (a) = a 0 , 1 a a a a1 a2 a2 2 2 a1 a a a a4 a2 a3 a4 , a1 a2 a 1 , 1 0 , a a4 a3 a3 , a3 a ~ ~ where a corresponds to each vi , (i = 1, 2, 3). In addition, we assume also that the fuzzy unmbers are given by the following values: ~ ~ ~ v1 = (2, 4, 6, 8 ), v 2 = (0, 3, 5, 7), v 3 = (18, 20, 22, 24). Setting = * = 0, then we get ~ v1 2 8, 0 ~ v2 7, 18 ~ v3 24. * * By choosing * ( 1 , * , 3 ) = (8, 7, 24), then the aspiration levels of 2 th e goals have been found h1= 10 and h2 = 15, respectively. The integer optimizati on problem associated with the first goal is P1 ( r ) : Minimize subject to L1= d 1 + d 1

2x1 + x2 + d 1 – d 1 = 10 x1 + x2 8 –x1 + x2 7 6 x1 + 2 x2 24 d1 0 , d1 0 , x1 0, x2 0 and integers.

448 O.M. Saad r The maximum degree of attainment of problem P1 ( the optimal integer solution: ) is L* 0 with 1 x1 =(2, 6) and d 1 0 , d 1 0 . The attainment problem for goal 2 is equivalent to the integer optimization prob lem P2 ( r ) where P2 ( r ): Minimize subject to L2= d 2 + d 2 x1 + 2x2 + d 2 – d 2 =15 2x1 + x2 + d 1 – d 1 = 10 d1 + d1 = 0 x1 + x2 8 –x1 +x2 7 6x1 + 2x2 24 d i 0 , d i 0 , x1 0, x2 0, and integers (i = 1, 2). The maximum degree of attainment of goal 2 is L*2 = 1 with the optimal integer s olution: x2 =(2, 6) and d 2 = 1, d 2 = 0 Therefore, the optimal integer solution of the original integer linear goal prog ram is: x* = (2, 6) L* 0 1 L2 = * with d 1 = 0, d 1 = 0 1 with d 2 = 1, d 2 = 0 with the corresponding used Gomory cut: x2 On the other hand, setting 4 ~ v1 =

* 7. = 1, we get: 5, 20 ~ v3 6, 3 ~ v2 22. * * * Choosing ( 1 , * , 3 ) = (6, 5, 22), then the optimal integer 2 solution o f the original program has been found:

Goal Programming For Fuzzy Integer MCDM 449 x* = (2, 4) L* 0 1 L2 = 1 * with with d 1 = 0, d 1 = 0 d 2 = 1, d 2 = 0 x2 10. with the corresponding used Gomory cut: 3x1 [0, 1] Remark. It should be noted that a systematic variation of will yield a ne w optimal integer solution to the integer linear goal program ( FILGP )~ v 10. CONCLUSION Since goal programming now encompasses any linear, integer, zero one, or nonline ar multi-objective problem (for which preemptive priorities may be established), the field of applications is wide open. The recent increase in interest in this area has already led to a large number of and wide variety of actual and propos ed applications. In this chapter, we have given numerical examples for the IMCDM problem and the FIMCDM. Fuzzy goal programming has many opportunities to develo p new approaches to it. REFERENCES Chanes, A., and Nilhaus, R.J., 1968, A goal programming model for manpower plann ing, Management Science Research Report, 115, Carnegie-Mellon University, Pittsb urgh, PA. Chankong, V. and Haims, Y.Y., 1983, Multiobjective Decision Making: (T heory and Methodology), Series Vol. 8, North Holland, New York. Charnes, A., and Cooper, W.W., 1961, Management Models and Industrial Applications of Linear Pro gramming, Wiley, New York. Chranes. A., et al., 1968, A Goal Programming Model f or Media Planning, Management Sciences, 138 151. Cohon, J.L., 1978, Multiobjecti ve Programming and Planning, Academic Press, New York. Dauer, J.P. and Osman, M. S.A., 1981, A Parametric Programming Algorithm for the Solution of Goal Programs with Application to Aggregate Planning of Production and Work Force, Technical Report, University of Nebraska-Lincoln, Lincoln, NE. Dauer, J.P., and Krueger, R .J., 1977, An iterative approach to goal programming, Operational Research Quart erly, 28: 671 681. Dubois, D., and Prade, H., 1980, Fuzzy Sets and Systems: Theo ry and Applications, Academic Press, New York.

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GREY FUZZY MULTI-OBJECTIVE OPTIMIZATION P.P. Mujumdar and Subhankar Karmakar Department of Civil Engineering, Indian Institute of Science, Bangalore, India Abstract: This chapter provides a description of grey fuzzy multi-objective optimization. A prerequisite background on grey systems, along with preliminary definitions is provided. Formulation of the grey fuzzy optimization starting with a general fu zzy optimization problem is discussed. Extension of the grey fuzzy optimization with the acceptability index to include multiple objectives is presented. Applic ation of the grey fuzzy multi-objective optimization is demonstrated with the pr oblem of waste load allocation in the field of environmental engineering. Grey f uzzy, fuzzy optimization, waste load allocation Key words: 1. INTRODUCTION Uncertainties in decision models may stem from a number of factors such as rando mness of input parameters, imprecision in management goals, inappropriateness in model selection leading to scenario uncertainty, broad range of possible altern ative formulations, and uncertainties in input parameters due to inadequate of d ata. Uncertainty due to randomness of input parameters may be modeled using the probability theory when adequate data are available to satisfactorily estimate t he probability distributions of the parameters. Uncertainties due to imprecision in the management problem, on the other hand, are modeled using the fuzzy sets theory, by appropriately constructing membership functions for the fuzzy or impr ecisely defined goals and constraints. In addition, model parameters in most opt imization problems need to be addressed as grey parameters, due to inadequate da ta for an accurate estimation but with known extreme C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 453

454 P.P. Mujumdar and S. Karmakar bounds of the parameter values. Such grey uncertainty, with partially known and partially unknown characteristics cannot be effectively modeled by probabilistic or fuzzy logic approach because of inadequacy of data to estimate probability d istribution and lack of information to precisely define the membership functions . Interval programming (IP) provides a methodology for modeling inexactness in p arameters (e.g., left-hand side model coefficients and righthand side stipulatio ns of constraints) of an optimization model, by considering them as interval num bers (Dantzig, 1963; Jaulin et al., 2001; Moore, 1979; Tong, 1994). A reason for the lack of many useful applications of interval programming is that the soluti on procedure is too complicated and time consuming (Dantzig, 1963; Jansson, 1988 ; Moore, 1979). Grey optimization (Huang et al., 1992, 1995, 2001) of grey syste ms theory (Deng, 1982) offers methods for incorporating uncertainties in model p arameters directly in an optimization framework avoiding huge data requirement a nd mathematical complicacy. The grey uncertainty or inexactness of model paramet ers can be addressed by representing them as interval grey numbers, instead of d eterministic real numbers. An interval grey number is a closed interval with kno wn lower and upper bounds but unknown distribution information (Huang et al., 19 92; 1995; Liu and Lin, 1998). Both interval and grey programming techniques are used for determining interval-valued solutions of an optimization model in which coefficients in objective function, left-hand side model coefficients, and righ t-hand side stipulations of constraints are represented by closed intervals. A b asic difference between interval programming and grey programming lies in their solution procedures. The primary goal of a grey optimization model is to determi ne the two extreme values of the optimal interval-valued decision variables in m ost adverse and favorable conditions (Huang et al., 1995, Karmakar and Mujumdar, 2005b). Huang et al. (1992, 1993, 1995), and Chen and Huang (2001) have present ed a few such novel efforts to find the solutions from grey linear, integer and quadratic programming models. A number of research contributions are available i n the literature that deal with uncertainty due to imprecision, fuzziness, or va gueness, where fuzzy sets theory (Zadeh, 1965) is the only tool used to address such uncertainty. The imprecision associated with management goals and constrain ts is quantified using membership functions, which are normally represented by a geometric shape that defines how each point in the input space is mapped to a m embership value between 0 and 1. For example, to account for the imprecision in the standards for determining a failure of water quality, occurrence of failure is treated as a fuzzy event (Mujumdar

Grey Fuzzy Multi-objective Optimization 455 and Sasikumar, 2002); a fuzzy set of low water quality maps all water quality le vels to “low water quality,” and its membership function denotes the degree to which the water quality is low. The membership functions represent the perceptions of the decision makers and other stakeholders in most decision-making problems. Th e boundaries (Ross, 1995) of the membership functions—or, the membership parameter s—are assumed fixed, and values to the parameters are assigned based on experience and judgment. As the model solution is likely to vary considerably with change in the membership functions, uncertainty in the boundaries and shape of the memb ership functions should also be addressed in a fuzzy optimization model. Some st udies address modeling of uncertainty in the values of membership parameters by considering the membership function itself as fuzzy. Type 2 fuzzy sets (Karnik a nd Mendel, 2001; Mizumoto and Tanaka, 1976; Mendel 2001; Zadeh, 1975), intervalvalued fuzzy sets (Chiang, 2001; Turksen and Bilgiç, 1996), and grey fuzzy optimiz ation (Chang et al., 1996; 1997; Karmakar and Mujumdar, 2005a, b; Maqsood et al. , 2005) are some examples of attempts to model such uncertainty. Mathematically grey fuzzy optimization is the simplest way to model the uncertainty in membersh ip parameters. In this approach, the membership parameters are considered as int erval grey numbers. A set of optimal interval-valued decision variables are obta ined as solution, corresponding to a maximized interval-valued goal fulfillment level, whereas conventional fuzzy optimization model gives only a single set of optimal decision variables corresponding to the maximum goal fulfillment level ( Chang et al., 1997; Karmakar and Mujumdar, 2005b; Zhang and Huang, 1994). This f eature of the solution from a grey fuzzy optimization model imparts flexibility in decision making. The width of the interval-valued optimal decision variables plays an important role in the grey fuzzy optimization model, as more width in t he optimal values of decision variables implies a wider choice to the decision-m akers. The width of the optimal interval-valued goal fulfillment level, on the o ther hand, implies the system uncertainty, which should be minimized in a grey f uzzy optimization model. Grey fuzzy multi-objective optimization is a potential approach to maximize the width of the interval-valued optimal decision variables for providing latitude in decision making and to minimize the width of the goal fulfillment level for reducing the system uncertainty (Karmakar and Mujumdar, 2 005a). The discussion in this chapter is restricted to grey fuzzy optimization t echniques mainly focusing on grey fuzzy multi-objective optimization. As a prere quisite, a brief overview of the grey systems theory is provided first.

456 P.P. Mujumdar and S. Karmakar 2. GREY SYSTEMS THEORY Grey systems theory was first proposed by Deng (1982). Concepts of grey systems are different from those of probability and statistics, which address problems w ith samples of a reasonable size, and also different from those of fuzzy mathema tics, which deal with problems with cognitive uncertainty. Table 1 presents some features of grey systems theory, probability theory, and fuzzy mathematics. Table 1. Features of Grey Systems Theory, Probability Theory and Fuzzy Mathemati cs (Deng 1982, Liu and Lin, 1998) Grey systems theory Small sample uncertainty I nformation coverage Few data points Any distribution Laws of reality Probability theory Large sample uncertainty Probability distribution Large number of data p oints Probability distribution Laws of statistics Fuzzy mathematics Cognitive un certainty Function of affiliation Experience Membership function Cognitive expre ssions Intention Foundation Characteristics Requirement Objective A grey system is a system other than a white system (system with completely know n information) and a black system (system with completely unknown information), and thus it has partially known and partially unknown characteristics. Table 2 s hows the major characteristics of the white, black, and grey systems. Table 2. Characteristics of White, Black and Grey Systems (Liu and Lin 1998) Whi te system Known Bright Order Surety Unique solution Black system Unknown Dark Ch aos Indulgence No result Grey system Incomplete Grey Complexity Tolerance Multip le solution Information Appearance Property Attitude Conclusion Most processes of interest in decision problems are in the grey stage due to the inadequate and/or fuzzy information. Grey fuzzy optimization provides a useful tool for decision making addressing such uncertainties. As a background to formu lation of a grey fuzzy optimization model, we

Grey Fuzzy Multi-objective Optimization 457 first provide a few definitions related to interval analysis (Moore, 1979) and g rey systems theory (Huang et al., 1995; Liu and Lin, 1998). DEFINITION 1. A “grey number” is such a number whose exact value is unknown, but a range within which th e value lies is known [Liu and Lin 1998]. Let x denote a closed and bounded set of real numbers. A grey number (x ) is defined as an interval with known lower ( x ) and upper (x ) bounds but with unknown distribution information for x (Huang et al., 1997). x = [x , x+] = [t x x t x] (1) x becomes a “deterministic number” or “white number” when, x = x = x . When x = [x , x+] = ( , + ) or x = [x1 , x2 ], that is, x has neither lower limit nor upper limit , or the lower and the upper limits are all grey numbers, x is called a “black num ber.” An “interval number” or “interval grey number” (x = [x , x+]) is one among several c lasses of grey numbers. DEFINITION 2. The “whitened value” of a grey number, x , is defined as a deterministic number with its value lying between the upper and low er xv x , when xv is a whitened value of x . For a bounds of x ; i.e., x general interval grey number, xv = x + (1 )x , [0, 1] (2) is called “equal weight whitenization” (Liu and Lin, 1998). “ ” is a weight factor that can take any value between 0 and 1. DEFINITION 3. In an equal weight whitenizati on, the whitened value obtained, when taking = ½, is called an “equal weight mean wh itenization” or “Whitened Mid Value” (WMV). Therefore, WMV of x is written as (Liu and Lin, 1998): xm = ½ (x + x ) (3) DEFINITION 4. The “grey degree” is a measure, useful for quantitatively evaluating t he quality of input or output uncertain information for mathematical models

458 P.P. Mujumdar and S. Karmakar (Huang et al., 1997). The “grey degree” of an interval grey number is defined as its width [x = (x – x )] divided by its WMV [xm = ½ (x + x )] (Huang et al., 1995) and is expressed in percentage (%) as follows: Gd(x ) = (x xm) 100 (4) where Gd(x ) is the grey degree of x . Solutions (model outputs) with considerab ly high grey degree have high width (x ) of output variables, which are consider ed as less useful and of poor quality for decision making. As the grey degree of objective function of an optimization model decreases, implying decreasing syst em uncertainties, the usefulness of the grey model increases. DEFINITION 5. A “gre y system” is defined as a system containing information presented as grey numbers [Huang et al., 1995]. DEFINITION 6. Let { , , , } be a binary operation on grey numbers. Therefore, the operations can be expressed as (Huang et al., 1995; Ishi buchi and Tanaka, 1990) x y = [min (x y), max (x y)], x x x ,y y y (5) For different binary operations: x + y = [(x + y ), (x + y )] x x x y = [(x y = [min (x y ), (x y )] y)] y (6) (7) (8) (9) y), max (x y = [min (x y), max (x y)], when 0 DEFINITION 7. A “general mathematical model” of grey linear programming is as follow s (Huang et al., 1992): Maximize f = c x subject to (10)

Grey Fuzzy Multi-objective Optimization 459 A x x 0 b (11) (12) where c = [c1 , c2 , ….., cn ]; x = [x1 , x2 , ….., xn ]T; b = [b1 , b2 , ….., bm ]T; A = {aij }, i = 1, 2,……, m, and j = 1, 2,……, n. Since interval grey parameters exist in the objective function and constraints, the f f optimal solutions of grey linear programming model are f = [ ˆ , ˆ ], and x = [ ˆ 1 , ˆ 2 , . . . . . , ˆ n ], where ˆ j = ˆ j , ˆ j ], x x x j = 1, 2,……, n. The x x x primary goal of a grey optimization model is to determine the two extreme x values of the optimal interval-valued decision variables, ˆ , in most adverse and favorable conditions considering the appropria te extreme bounds of the pre-specified parameters in the model constraints, i.e. , c , A , b . It is to be noted that the grey optimization model does not includ e the situation when a model parameter expressed as an interval grey number, con tains a zero with the two bounds having different signs (e.g., bi = [ bi-, +bi+] , where bi>0). Details of the solution algorithm for a grey optimization problem may be found in Huang et al. (1994, 1995). 3. CONCEPT OF A GREY FUZZY DECISION Fuzzy optimization (Zimmermann, 1978) is an application of fuzzy sets theory tha t determines optimal solution in the presence of imprecisely defined goals and c onstraints. Bellman and Zadeh (1970) proposed a broad definition of the fuzzy de cision as a confluence of fuzzy goals and fuzzy constraints, which is the basis of fuzzy optimization. Noting that the decision space is defined by the intersec tion of different fuzzy goals, the fuzzy decision D is written as follows: D F1 F2 (13) where fuzzy sets F1 and F2 represent the two fuzzy goals. The membership functio n of the fuzzy decision (D) is given by D ( x) min [ F1 ( x ), F2 ( x)]. (14)

460 P.P. Mujumdar and S. Karmakar “ ” is the measuring variable corresponding to the membership function of fuzzy deci sion (D), which reflects the degree of fulfillment of the system goals. A termin ology of “goal fulfillment level” is used throughout the chapter to represent “ .” In th e concept of fuzzy decision (D) as described by Eq. (13), the arguments of F1 an d F2 are deterministic real numbers (x). When the goals F1 and F2 are imprecise fuzzy goals or grey fuzzy goals, i.e., the uncertain membership parameters are c onsidered as interval grey numbers and corresponding arguments are interval grey numbers ( x ), the fuzzy decision leads to a “grey fuzzy decision (D )” (Karmakar a nd Mujumdar, 2005a, 2000). This terminology is earlier used by Luo et al. (1999) to define a “grey fuzzy motion decision” combining grey prediction and fuzzy logic control theories. The notion of “grey fuzzy decision” presented in this chapter is d ifferent from that used by Luo et al. (1999). Here grey fuzzy decision represent s the fuzzy decision resulting from the imprecise membership functions, where th e membership parameters are expressed as interval grey numbers (Karmakar and Muj umdar, 2005b). Figure 1 illustrates the concept of grey fuzzy decision consideri ng the confluence of two imprecise membership functions for grey fuzzy goals, F1 and F2 . Considering “logical and,” corresponding to the “set theoretic intersection” a s an aggregation operator, the grey fuzzy decision is determined. In Figure 1, t he decision space is is defined by the lower and upper boundaries A “FNGH” and A“ECMC’HH ,” respectively. Goal F 1 1 A B B’ D Goal F2 D’ D’’ M C N A’’ E F C’ 0 G H H’’ x Upper bound of imprecise membership function Lower bound of imprecise membership function Figure 1. Concept of grey fuzzy decision

Grey Fuzzy Multi-objective Optimization 461 The solutions ˆ , corresponding to the maximum value of the x membership function of the resulting grey fuzzy decision (D ) is an interval in the space CMC’N (Figur e 1). Mathematically the grey fuzzy decision (D ) for F1 and F2 can be defined w ith the imprecise membership function (Karmakar and Mujumdar, 2005a): D (x ) min [ min{ F1 ( x ), F2 ( x )}; min{ F1 ( x ), F2 ( x )}] (15) (16) D (x ) max [ min{ F1 ( x ), F2 ( x )}; min{ F1 ( x ), F2 ( x )}] where D (x )

and D (x ) are lower and upper bounds of the imprecise membership functions for an interval [x–, x+], respectively. Eqs. (15) a nd (16) are valid for all combinations of imprecise membership functions (i.e., non-increasing, nondecreasing, or a combination of the two). Eqs. (15) and (16) are readily extendible to any number of imprecise goals. 4. GREY FUZZY OPTIMIZATION The grey fuzzy optimization technique is based on the concept of grey sion. Determination of an appropriate deterministic equivalent of the optimization model is still a potential research area. Following the used by Zimmermann (1985), a generalized fuzzy optimization model may as Maximize ( subject to 0 x 1 0, x ) di di n Bi x di i i, for i 1, .... , m (17–20) The solution of model (17) (20) gives the optimal values of x satisfying all m n umbers of fuzzy goals, expressed by the constraint Bix di for i = 1,…, m; ~ fuzzy deci grey fuzzy notations be written

462 P.P. Mujumdar and S. Karmakar with maximized level of goal fulfillment, ˆ . Here “ ~” is “fuzzified” version of “ ” and has the linguistic interpretation “essentially less than or equal t o.” Constraint (18) denotes the ith membership function, i(x), interpreted as the degree to which x fulfills the fuzzy goal, where Bi and di denote the i-th row o f B and d, respectively. The exponent i is a nonzero positive real number. Assig nment of numerical value to this exponent is subject to the desired shape of the membership functions. A value of i = 1 leads to the linear membership function. The value of i(x) should be 0 if the set of constraints are strongly violated a nd 1 if they are well satisfied. i(x) should increase monotonically from 0 to 1. Figure 2 shows a linear membership function of Bix, where two membership parame ters are at the desirable (di) level and maximum permissible [di’ = (di + pi)] lev el, fixed by the decision maker. 1 Slope = (1/pi) 0 di di’ =(di+ pi) Bix Figure 2. Linear membership function The value of pi is uncertain and a subjectively chosen constant of the admissibl e violation for i-th fuzzy goal. The membership function, i(x), results in an im precise membership function, i ( x ) , when the uncertainty in the value of pi i s considered as interval grey number ( pi ). The current discussion focuses on m odeling the uncertainty in membership parameters considering the boundaries of t he membership functions as interval grey numbers, which results in the value of pi as an interval grey number. Figure 3 shows a linear imprecise membership func tion where the uncertain value of pi is expressed as ( d i d i ) , and extreme b ounds are presented as pi ( d i d i ) and pi ( d i d i ) .

Grey Fuzzy Multi-objective Optimization i 463 1 Slope = [1/(di’ - di)] 0 di d i di di d i’ di Bi x Figure 3. Linear imprecise membership function Mathematically the imprecise membership function expressed as i (x ) can be 1 i if i Bi x di Bi x di di for i 1, ... , m (21–23) (x ) di Bi x di di 0 if di if Bi x Similar to the max min formulation for fuzzy optimization by Zimmermann (1978), the grey fuzzy optimization model can be represented as Find (x ) maxmin x x 0, 0 i di Bi x di di i (24–25) In the grey fuzzy optimization model, the input vector Bi can also be uncertain, depending on the particular problem being solved. A more generalized form of th e grey fuzzy optimization model can be obtained by considering Bi as an interval grey number ( Bi ). Similar to fuzzy optimization model (17) (20), a generalize d form of the grey fuzzy optimization model may be written as

464 P.P. Mujumdar and S. Karmakar Maximize ( subject to 0 x 0 1 di ) ( Bi x) di di i i (26–29) where Bi is an interval grey number that results in the value of (Bix) [i.e., x ) ] as an interval grey number, following Eq. (8) in Definition 6. (Bi A typical confluence of two non increasing linear imprecise membership functions as descri bed in Eqs. (21) (23), when i = 1, 2 and i = 1, are shown in Figure 4. In Figure 4, the lower and upper boundaries of grey fuzzy decision are ABD’F’FG and ABCDEFG, respectively. 1 A B C Gray fuzzy goal 1 D D’ E Gray fuzzy goal 2 0 F’ F G d1 d1d1 d2 d2 d2 d2 d2 d2 d1 d1 d1 (Bx) Figure 4. Confluence of gray fuzzy goals To obtain the two extreme values of optimum goal fulfillment level ( ˆ and ˆ ), that provide solutions for two extreme cases encompassing all intermediate possibili ties, the deterministic equivalent of the grey fuzzy optimization model [Eqs. (2 6) (29)] is divided into two sub-models as:

¢

Grey Fuzzy Multi-objective Optimization 465 Sub-model 1 Maximize ( subject to ) di 0 x 0 ( Bi x) di di 1 i i (30–33) Sub-model 2 Maximize ( subject to ) di ( Bi x) di di i i (34–38) x 0 x ˆ x from submodel 1 1 0 Sub-model 1 is formulated to obtain the upper bound of a maximized minimum goal fulfillment level ( ˆ ) and the corresponding optimal value x of the decision vari able ( ˆ ). The left-hand side (LHS) of constraint (31) is written considering the maximum possible values of the LHS of Eq. (27). The maximum possible value of t he LHS of Eq. (27) occurs with the numerator taking the highest value and the de nominator, the lowest. Using the same argument as in Sub-model 1, Sub-model 2 [( 34) (38)] is formulated to obtain the lower bound of the maximized minimum goal fulfillment level ( ˆ ) and corresponding optimal value of decision variable x ( ˆ ) . The LHS of constraint (35) is written considering the minimum possible values of Eq. (27). The minimum possible value of the left-hand side of Eq. (27) occurs with the numerator taking the lowest value and the denominator, the highest. To ensure that the optimal upper bound of the x decision variable ˆ , obtained from Sub-model 2 is at least equal to the x optimal lower bound of the decision varia ble ˆ , obtained from the Submodel 1, an interactive constraint (36) (Huang et al. , 1995) is added. When

466 P.P. Mujumdar and S. Karmakar the problem is complex and many decision variables with functional relationships are present, a direct comparison of the dominance of x+ or x– , or , is i.e., whe ther x+ or x- corresponds to maximized value of impossible to know prior to solv ing the models. The appropriateness of submodel formulation on finding out suita ble deterministic equivalent of the grey fuzzy optimization model depends on the values of intervalvalued membership parameters and the consequent intersection of the grey fuzzy goals (Karmakar and Mujumdar, 2005b). For a given set of inter valvalued membership parameters, if a particular formulation represents an appro priate deterministic equivalent, other alternative formulations do not. The appr opriate deterministic equivalent of a grey optimization model should give the lo west value of grey degree of . The solution approach for the fuzzy optimization problem using the max min operator (Zimmermann, 1978) may not result in a unique solution (Dubois and Fortemps, 1999; Lai and Hwang, 1992; Lin, 2004). To impart flexibility in decision making, the multiple solutions of the fuzzy optimizatio n model may be obtained as a parametric equation or equations that represent a s ubspace. Determination of such a subspace in a fuzzy optimization problem is its elf a potential research area (Lai and Hwang, 1992; Li, 1990; Lin, 2004). It is also observed that as the number of objectives and decision variables increases in the fuzzy optimization model, the possibility of existence of multiple soluti on increases. When the deterministic equivalents of the grey fuzzy optimization model lead to fuzzy optimization models with a max min operator, therefore, atte ntion must be given to multiple solutions. Solutions from the grey fuzzy optimiz ation model enhance the flexibility and applicability in decision x x making, as the decision maker gets a range of optimal solutions, [ ˆ , ˆ ] . The width of the interval-valued solutions thus plays an important role in the grey fuzzy optimiz ation model. The grey fuzzy multi-objective optimization technique discussed in the next section maximizes the width of the interval-valued decision variables, ( x x ), (Huang and Loucks, 2000; Karmakar and Mujumdar, 2005a) in a multi-objec tive framework. Similar to the grey fuzzy optimization model, the upper and lowe r bounds and ) are maximized in the grey of the goal fulfillment level (i.e., fu zzy multi-objective optimization technique, but additionally the width of ) is a lso minimized, thus the degree of goal fulfillment level, ( reducing the system uncertainty.

Grey Fuzzy Multi-objective Optimization 467 5. GREY FUZZY MULTI-OBJECTIVE OPTIMIZATION

The grey fuzzy optimization model given in Eqs. (26) (29) forms the basis of the grey fuzzy multi-objective optimization technique. The inequality constraint (2 7) addresses the grey fuzzy management goals in the optimization model. The cons traint set (27) defines the order relations (e.g., the relations “greater than or equal to” or “less than or equal to”) containing interval grey numbers on both sides. Determination of meaningful ranking between two partially or fully overlapping i ntervals in the order relations is a potential research area (e.g., Ishibuchi an d Tanaka, 1990; Moore, 1979; Sengupta et al., 2001). Recently, Sengupta et al., (2001) proposed a satisfactory deterministic equivalent form of inequality const raints containing interval grey numbers by using the acceptability index (A). Th e acceptability index (A) is defined as the grade of acceptability of the premis e that the “first interval grey number (a ) is inferior to the second (b ),” denoted as a (<) b . Here, the term “inferior to” (“superior to”) is analogous to “less than” (“gr er than”). The acceptability index (A) is expressed as (Sengupta et al., 2001) A [ a (<) b ] = [m(b ) m(a )] /[ w(b ) w(a )] (39) where [w(b ) + w(a )] 0; w(a ) is the half-width of a = ½ (a a ); m(a ) is the mea n of a = ½ (a + a ). Notations are similarly defined for the interval grey number b . The grade of acceptability of a (<) b may be classified and interpreted furt her on the basis of the comparative position of mean and the half-width of inter val b with respect to those of interval b , where y is a a . Let us consider an interval inequality relation a y deterministic variable. A satisfactory determin istic equivalent form of b , is proposed as (Sengupta et al., interval inequalit y relation a y 2001): a y b {a y b and A [ a y (<) b ] [0, 1]} (40) where is interpreted as an optimistic threshold fixed by the decision maker. Sim ilarly, a satisfactory deterministic equivalent form of interval inequality rela tion a y b is proposed as (Sengupta et al., 2001): a y b {a y b and A [ a y(>) b ] [0, 1]} (41)

¢

¢

468 P.P. Mujumdar and S. Karmakar where the symbol (>) indicates “superior to,” which is analogous to “greater than.” The deterministic equivalent of the grey fuzzy optimization model given in Eqs. (26) (29) is formulated using the expression (40). By using the attributes mean, wid th, and acceptability index of the interval grey numbers, the grey fuzzy optimiz ation model is reduced to a deterministic multi-objective optimization model, as follows: Maximize Maximize Minimize [( subject to i ) /( ( Bi x) }/( d i )] d i )] i (42–45) (x ) [{d i A [ { di ( Bi x ) } /( d i d i ) (<) ] i [0, 1] (46) x x 0 x 0, x 0 1, 0 1 (47–50) The constraints (45) (46) together define the deterministic equivalent of the co nstraint (27). The acceptability index in constraint (46) compares the interval grey numbers in the inequality constraints (27). In constraints (46), i is the o ptimistic threshold for the i-th constraint fixed by the decision maker. In this model, the grey fuzzy management goal as expressed by constraint (27), is repre sented by linear imprecise membership functions [i.e., substituting i = 1 in con straint (27)] as the Eq. (40) with acceptability index for ranking the interval grey numbers in the inequality constraints is applicable only for linear program ming problems (Sengupta et al., 2001). The expression (39) of the acceptability index may be substituted in the constraint (46), to obtain a simplified form wit h algebraic operations on the interval grey numbers (Liu and Lin, 1998; Moore, 1 979). The objectives (42) and (43) maximize the upper and lower bound of the goa l ), respectively, which ensure the maximum fulfillment level ( possibility of f ulfillment of the grey fuzzy goal. The objective (44)

Grey Fuzzy Multi-objective Optimization 469 minimizes the width of the goal fulfillment level [ w ( ) ] with maximization of the denominator, ( ) , to be consistent with the first two objectives, (42) and (43). This objective is included as the reduction of the width of the goal fulf illment level implies reduction in system uncertainties and an increase in effec tiveness of the grey model (Huang et al., 1995). Similarly, a higher flexibility (i.e., higher width of the interval) of the decision variables ( x ) is always desirable, as it allows a wider choice to the decision-makers. The objectives (4 2) – (44) do not address the maximization of the width of decision variables [i.e. , ( x x )]. The width of the decision variables may be maximized along with the objectives (42)–(44) while solving the multi-objective optimization model [(42)–(50) ] using the fuzzy multi-objective optimization technique (Sakawa, 1984) or the f uzzy goal programming technique (Pal and Moitra, 2003; Sakawa et al., 1987). The procedure of solution is discussed through an application in the next section. 5.1 An Application in Environmental Engineering A number of successful applications of grey systems theory have been found in ma ny areas of human endeavors, including agriculture, transportation, hydrology, e nvironment, economics, water resources systems, and control theory. Table 3 show s some recent applications of grey systems theory in the field of environmental and water resources engineering. An application of grey fuzzy multi-objective op timization technique is demonstrated with a waste load allocation problem here. Waste load allocation (WLA) in a stream refers to the determination of required treatment levels of pollutants (fractional removal levels) [e.g., biochemical ox ygen demand (BOD) loading, toxic pollutant concentration, etc.] at a set of poin t sources of pollution to ensure that water quality is maintained at desired lev els throughout the stream. A common practice of the pollution control agency (PC A) to ensure an acceptable water quality condition is to check the water quality at a finite number of locations in the river. These locations are called water quality checkpoints. A WLA model for decision making in water quality control in a river system, in general, integrates a water quality simulation model, measur ing the influence of a pollutant on a water quality indicator [e.g., dissolved o xygen (DO) deficit, hardness, nitrate-nitrogen concentration, etc.] at a downstr eam location with an optimization model to provide best compromise solutions acc eptable to both PCA and dischargers (e.g., municipal and industrial

470 P.P. Mujumdar and S. Karmakar Table 3. Applications of Grey Systems Theory in Water Resources and Environmenta l Engineering Application Literature Case study Parameters considered as interva l grey numbers Hypothetical Existing landfill capacity, Municipal solid Chen and Huang treatment capacity, generated (2001), Huang et al. study area waste resid ues after treatment, management and (1992, 1993), Zou revenue from waste-toet al . (2000) planning energy facility. Water quantity & quality management Huang et al. (1996), Huang and Loucks (2000) 1. Fujian province of China, 2. Hypothetical study area Water quantity: crop water requirement, municipal water requirement, cost for obtaining-transportingdelivering-allocating water, cost of manure coll ection, cost of fertilizer application, average returns from livestocks, etc.; W ater quality: amount of manure generated by humans, livestocks, amount of manure applied to soil, population in study area, number of livestock, nitrogen volati lization and denitrification rates, area under the crops, pollutant losses, etc. Rainfall forecasting Yu et al. (2000) Areal mean rainfall. San-Hsia and Heng-Chi subcatchments in Tahan creek, Taiwan Shiman reservoir in Taiwan Storage of the upper and lower curves in rule curves at i-th stage, inflows, supply for irrigation, municipal and industrial purposes . Degree of aspiration levels, BOD loading, construction and average operating c ost of treatment plants, removal efficiency of BOD, deoxygenation and reaeration coefficients. Reservoir operation Chang et al. (2002) Water quality Chang et al. (1997), control problems Wu et al. (1997), (rivers an d lakes) Karmakar and Mujumdar (2005a, 2005b) 1. Tseng-Wen river basin in south Taiwan, 2. Lake Erhai in southwestern China Coastal waste water treatment Chang and Wang (1995) Guishuic waste Concentrations of water treatment conservative pollutants, projec t in Taiwan waste water flow rates, initial dilution, length of diffusers.

Grey Fuzzy Multi-objective Optimization 471 dischargers). A number of WLA models have been developed in the past for optimal allocation of assimilative capacity of a river system considering uncertainties due to randomness in input variables (e.g., stream flow, effluent flow, tempera ture, reaction rates, etc.) and imprecision in management goals (e.g., goals of PCA and dischargers), the latter being addressed using the fuzzy sets theory. Im precision in management goals is usually modeled using fuzzy membership function s, specifying the desirable maximum permissible levels of the goals by prespecif ied membership parameters. Choice of appropriate values of membership parameters is an important issue in any fuzzy optimization model, as these are highly subj ective. In a water quality control problem, such subjectivity in choice of param eters results in an uncertainty in the membership parameters and leads to a seco nd level of fuzziness in the model, with the membership functions themselves bei ng imprecisely stated. Moreover, in practical situations, for the same water qua lity indicator, different water quality standards are used for different uses, w hich results in an uncertainty in the membership parameters of the goals of PCA. Two sets of conflicting goals associated with the river water quality managemen t are generally considered in a waste load allocation problem. The PCA specifies the desirable concentration level (cDjl) and maximum permissible concentration level (cHjl) of the water quality indicator j at the water quality checkpoint l (cDjl cHjl). The goal of the PCA (E jl ) is to make the concentration level (cjl ) of water quality indicator j at the checkpoint l as close as possible to the d esirable level, cDjl, so that the water quality at the checkpoint l is enhanced with respect to the water quality indicator j, for all j and l. This goal is rep resented by a membership function. For example, if the DO-deficit is the water q uality indicator, a non-increasing membership function suitably reflects the goa ls of the PCA with respect to DO-deficit at a checkpoint. The uncertainty associ ated with membership parameters (cDjl and cHjl) is addressed using interval grey numbers, and the membership parameters are expressed as cD jl and cH jl. Using nonincreasing imprecise membership functions, the grey fuzzy goals of PCA are ex pressed as 1 ( c jl ) jl c jl H cD jl c jl c jl H [( c jl 0 c jl ) /( c jl H c jl )] D jl c jl c jl D c jl

H (51) E

472 P.P. Mujumdar and S. Karmakar The exponent jl is a nonzero positive real number. Assignment of numerical value to this exponent is subject to the desired shape of the membership functions. A value of jl = 1 leads to a linear imprecise membership function. The grey fuzzy goals of the dischargers are similarly expressed as: 1 M [( x mn 0 M x jmn ) /( x mn L x mn )] jmn x jmn L x mn L x mn M x jmn x mn F jmn ( x jmn ) M x jmn x mn (52) where the aspiration level and the maximum acceptable level of fractional remova l of the pollutant n at discharger m are represented as xL mn and xM mn, respect ively (xL mn xM mn). Similar to the exponent jl in Eq. (51), jmn is a nonzero po sitive real number. The goal of the dischargers (F jmn ) is to make the fraction al removal level (x jmn) as close as possible to xL mn, to minimize the waste tr eatment cost for pollutant n. These two sets of conflicting grey fuzzy goals are incorporated in the optimization model using the grey fuzzy decision concept. U sing the concept of grey fuzzy optimization discussed in Section 4, the grey fuz zy waste load allocation model (GFWLAM) is written as (Karmakar and Mujumdar, 20 05b): Maximize subject (c E jl ( x jmn ) jmn c jl cH jl M x mn j ,l j ,m ,n M [( x mn M x jmn ) /( x mn L x mn )] jmn to jl ) [( c H jl c jl ) /( c H jl cD jl )] jl j ,l (53–58) j ,m ,n F cD jl L x mn 0 x jmn 1 The constraints (54) and (55) are constructed from imprecise membership function s for the grey fuzzy goals of PCA and dischargers, respectively. The crisp const raints (56) and (57) are based on the water quality requirements specified by th e PCA and possible fractional removal levels, respectively. Constraint (58) repr esents the bounds on the parameter . In the expression for goals of PCA [constra int (54)], the concentration

Grey Fuzzy Multi-objective Optimization 473 level c jl, of water quality indicator j at checkpoint l, may be mathematically expressed as: c jl f ( x jmn ) (59) where the transfer function f indicates the aggregated effect of all pollutants and dischargers (located upstream of checkpoint l) on the water quality indicato r j. The transfer function can be evaluated using appropriate mathematical model s that determine spatial distribution of the water quality indicator due to poll utant discharge into the river system from point sources (Sasikumar and Mujumdar , 1998; Mujumdar and Sasikumar, 2002). The fractional removal levels (x jmn) and the goal fulfillment level ( ) are the decision variables in this model. The gr ey fuzzy inequality constraints (54) and (55) addressing the goals of the PCA an d dischargers are the order relations containing interval grey numbers on both s ides. A satisfactory deterministic equivalent of these constraints can be obtain ed using the concept of acceptability index (A) as defined in Eq. (40). The dete rministic equivalent of the grey fuzzy optimization model (53) (58) can be formu lated using the methodology of formulating the multi-objective optimization mode l as presented in (42) (50) and expressed as Maximize Maximize Minimize [( subject to E jl (c jl ) [(c H jl M [( xmn c jl ) / (c H jl c D )] jl L xmn )] j, l j , m, n ) /( )] (60–64) F jmn ( x jmn ) M x jmn ) /( xmn A [( cH jl c jl ) /( c H jl c D (<) ) jl L x jmn ) (<) ] 1 [0, 1] [0, 1] j, l (65) (66)

A [( xM jmn M x jmn ) /( x jmn ] 2 j, m, n

474 cD jl L x mn c jl cH jl M x mn c jl x jmn 0 1; c ; ; jl cD jl L x mn c jl x jmn cH jl P.P. Mujumdar and S. Karmakar j,l j, m , n j,l j,m ,n 1 x jmn M x mn (67–72) x jmn 0 The constraints (63) (66) are the deterministic equivalent of the constraints (5 4) (55) using the acceptability index. The goals of the PCA and dischargers are represented by linear imprecise membership functions (i.e., jl, jmn = 1). The ob jective (62) minimizes the system uncertainty by minimizing [( + –)/( ++ –)]. In the multi-objective optimization model (60)–(72), the three objectives are optimized individually in three separate sub-problems along with the contraints (63) (72) to obtain the maximum and minimum possible values of +, – and [( +– –)/( ++ –)] [i.e., i deal points and worst possible values of the fuzzy multi-objective optimization technique (Sakawa, 1984)], respectively. As discussed earlier, another objective of the river water quality management is to permit more flexibility (i.e., more width of the interval) in the optimal fractional removal level ( ˆ jmn ). Thus, m aximization of the grey degree of x jmn is x considered as another objective alo ng with objectives (60) to (62). The maximum and minimum values of the grey degr ee of x jmn are determined from the three sub-problems. All the objectives are q uantified by using appropriate membership functions according to the fuzzy multi -objective optimization technique (Sakawa, 1984). The fuzzy decision concept wit h a “minimum” operator is applied to aggregate the membership functions of the objec tives (60) (62) along with other objectives for minimizing Gd (x jmn) for the di schargers. The solution algorithm for the problem (60)–(72) is as follows: (1) Sol ve three sub-problems, each formulated with one objective ( +, – and [( +– –)/( ++ – )]) and all constraints. (2) From the three sets of solutions, obtain the best and worst values of +, – , [( +– –)/( ++ –)] and Gd(x jmn). (3) Define membership functions for + – (non decreasing), [( +– –)/( ++ –)] (non (non-decreasing), increasing) and Gd(x jmn) (non decreasing) with their best and worst values. (4) Maximize the minimum membership of the objectives using the fuzzy decision concept with the max–min ap proach. This gives the

Grey Fuzzy Multi-objective Optimization 475 solution for the grey fuzzy multiobjective optimnization problem, Eqs. (60) (72) . Application of the grey fuzzy multi-objective optimization model [Eqs. (60) (7 2)] for water quality management is demonstrated on a hypothetical river system shown in Figure 5. LEGEND D1 1 3 1 2 D2 4 5 6 2 7 8 9 12 10 11 13 14 15 16 4 17 18 3 D3 D4 1 to 18 River flow Checkpoints e Dm River reach e Discharger m Figure 5. Hypothetical river system In this application, the water quality indicator of interest is the DOdeficit at 18 checkpoints in the river system due to the point sources of BOD from four di schargers. The saturation DO concentration is taken as 10 mg/L for all the reach es. A deterministic value of river flow of 7 Mcum/day is considered. The notatio ns of variables are simplified by retaining only the suffixes l (checkpoints) an d m (dischargers) in the model (60) (72) dropping the suffixes j and n as there is only one water quality indicator (DO deficit) and only one pollutant (BOD). D etails of the effluent flow and imprecise membership functions are given in Tabl es 4 and 5, respectively ( Data from Mujumdar and Sasikumar, 2002). Table 4. Effluent Flow Data Discharger 1 2 3 4 Effluent flow rate (104 m3/day) 2.134 6.321 7.554 5.180 BOD ( mg/L) 1250 1415 1040 935 DO (mg/L) 1.230 2.400 1.700 2.160

476 P.P. Mujumdar and S. Karmakar Table 5. Details of Imprecise Membership Functions* River reach 1 2 3 4 Checkpoints 1 2 3 6 7 11 12 18 cDl (mg/L) + (0.00) 0.00 (0.10) 0.00 (0.20) 0.17 (0.20) 0.17 0.00 0.10 0.22 0.22 cHl (mg/L) + (3.00 2.70 (3.00) 2.70 (3.50) 3.30 (3.50) 3.30 3.20 3.20 3.70 3.70 xLm + (0.30) 0.25 (0.30) 0.25 (0.35) 0.30 (0.35) 0.30 0.35 0.35 0.40 0.40 xMm + (0.85) 0.80 (0.85) 0.80 (0.85) 0.80 (0.85) 0.80 0.90 0.90 0.90 0.90 ( ): Deterministic values of membership parameters, “ ” : Lower bound, “+” : Upper bound , “ ”: Data from Karmakar and Mujumdar (2005a) In constraints (65) and (66), 1 and 2 are optimistic thresholds, which are set t o zero in the current application, and thus, a conservative optimal solution is obtained, implying a stringent restriction on water pollution. The decision make r selects the values of 1 and 2 equal to zero when the water quality management issues in the river system are too critical and important; otherwise some optimi stic strategy can be considered by choosing values of 1 and 2 close to unity. Fo r most water quality indicators, a high level of fractional removal of pollutant s (e.g., BOD loading, toxic pollutant concentration, etc.) results in a low leve l of water quality indicator (e.g., DO-deficit, nitrate-nitrogen concentration, etc.). The lower bound of water quality indicator (c–l) is therefore expressed in terms of the upper bound of fractional removal level (x+m) and similarly, c+l is expressed in terms of x–m, using the one-dimensional Streeter Phelps model for a BOD–DO relationship in a stream (Streeter and Phelps, 1925). Further, using the re cursive relationships given by Fugiwara et al., 1987; 1988), the DO-deficit is w ritten as a linear function of the fractional removal levels. This results in x jmn as the only decision variables in the optimization model (60) (72). Table 6 shows the expressions of the DOdeficit at the 18 checkpoints in terms of fractio nal removal levels of BOD waste load by dischargers, situated upstream of the pa rticular checkpoint. For example, the DO-deficit at checkpoint 15 is expressed a s follows using the data given in Table 6: c15 0.309 0.201 x1 0.079 x2 0.024 x3 0.002x4 (73)

Grey Fuzzy Multi-objective Optimization Table 6. DO-Deficit River reach 1 Checkpoints 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 5 16 17 18 Constant terms 0.1142 0.1935 0.2595 0.6198 0.9472 1.2433 1.3230 1.832 7 2.2948 2.7140 3.0919 3.1175 3.6006 4.0345 4.4259 4.7756 5.0877 5.3635 ( 1) Coe fficients of fractional removal levels 477 x1 0.0893 0.1702 0.1687 0.2409 0.3062 0.3651 0.3628 0.4149 0.4620 0.5041 0.5418 0.5 361 0.5691 0.5980 0.6238 0.6462 0.6656 0.6823 x2 — — — 0.2942 0.5618 0.8042 0.7992 1.0150 1.2095 1.3862 1.5453 1.5289 1.6690 1.7928 1.9 039 2.0023 2.0894 2.1652 x3 — — — — — — — 0.2524 0.4828 0.6923 0.8821 0.8734 1.0412 1.1937 1.3309 1.4538 1.5635 1.6611 x4 — — — — — — — — — — — — 0.1538 0.2935 0.4211 0.5366 0.6413 0.7354 2 3 4 Substituting the values of membership parameters and the expressions of DO-defic it in terms of BOD removal levels from Tables 5/6, respectively, in the grey fuz zy multi-objective optimization model (60)– (72) and solving the resulting linear programming problem, optimal interval-valued fractional removal levels of BOD ar e determined as presented in Table 7. In Table 7, columns 2–4 show the results obt ained from Sub-problems 1–3, i.e., maximization of +, maximization of – and minimiza tion of [( + –)/( + + –)], respectively. The minimum and maximum values of +, , [( +– –) / ( ++ –)] and Gd(x 1), … , Gd(x 4) are taken from the columns 2–4; rows 6, 5, 8, and 9–12, respectively. For example, columns 2–4, row 5, show the values of – obtained fr om Subproblem 1–3, respectively. The maximum value of – (i.e., 0.3121) is obtained f rom Sub-problem 2, and the minimum value (i.e., 0.0006) is obtained from Sub-pro blem 1. The requirements of all the objectives are quantified by defining linear membership functions with the minimum and maximum values of the objective funct ions as membership parameters. Column 5, rows 1 6, show the optimal fractional r emoval levels of the x pollutants by different dischargers ( ˆ ) and corresponding ˆ values.

478 Table 7. BOD Sl. No. Solution Sub(1) problem 1 (Max. +) (2) 1 X1 [0.4845, 0.7751 ] 2 X2 [0.4682, 0.7987] 3 X3 [0.5190, 0.7951] 4 X4 [0.5172, 0.7970] 5 0.0006 + 6 0.9592 7 Gd ( ) — 8 ( + - –) 0.9987 /( + + –) 9 Gd(x 1) 0.4615 10 Gd (x 2) 0.5217 11 Gd (x 3) 0.4202 12 Gd (x 4) 0.4259 13 Avg. — Gd (x ) Subproblem 2 (Max. ) (3) [0.5 955, 0.5964] [0.5967, 0.5970] [0.6124, 0.6126] [0.6121, 0.6127] 0.3121 0.3618 — 0. 0737 0.0015 0.0004 0.0003 0.0010 — Subproblem 3 [Min.( +– –) /( ++ –)] (4) [0.6060, 0.65 31] [0.4301, 0.6578] [0.5517, 0.5679] [0.6324, 0.6517] 0.1052 0.1052 — 0.0000 0.07 48 0.4186 0.0288 0.0300 — P.P. Mujumdar and S. Karmakar Multi- Deterministic objective model GFWLAM(6) (5) [0.5268, [0.6150, 0.6652] 0.6 150] [0.5302, [0.6150, 0.6656] 0.6150] [0.5367, [0.6360, 0.6757] 0.6360] [0.5357 , [0.6360, 0.6747] 0.6360] 0.2066 0.4277 0.5064 0.4277 0.8411 0.0000 — 0.2322 0.22 65 0.2293 0.2297 0.2294 — 0.0000 0.0000 0.0000 0.0000 0.0000 GFWLAM (7) [0.5970, 0.6410] [0.5970, 0.6410] [0.6120, 0.6700] [0.6120, 0.6700] 0.3126 0.574 5 0.5903 — 0.0711 0.0711 0.0905 0.0905 0.0812 To evaluate the quality of input or output uncertain information, a measure of “Gr ey degree” [Eq. (4)] is used. As the grey degree of the optimal value of the objec tive function decreases, the effectiveness of the grey model increases with decr easing system uncertainties. Substituting the deterministic values of membership parameters given in Table 5, in the grey fuzzy optimization model (53) (58), th e optimal fractional removal levels of BOD are determined as presented in column 6 of Table 7, for which average value of the grey degree of input parameters is zero. In column 7 of Table 7, the solutions obtained from GFWLAM based on the g rey fuzzy optimization technique (Section 4) are presented. For this solution, v alues of all input parameters are considered the same as those for the multi-obj ective GFWLAM. Comparing the results shown in column 5 and column 7 (rows 1–4) it may be concluded that the widths of x optimal fractional removal levels of BOD ( ˆ ) for multi-objective GFWLAM are more than those of GFWLAM because of the inclu sion of the objective of maximization of grey degrees of fractional removal leve ls. x The same observation can also be made by comparing the Gd( ˆ ) values shown in rows 9–12 of columns 5 and 7. The value of Gd( ˆ ) is, however,

Grey Fuzzy Multi-objective Optimization 479 more than the value obtained from GFWLAM, which indicates more uncertainty in th e system compared with that resulting from the GFWLAM solution. The result obtai ned from multi-objective GFWLAM is more useful to the decision makers as it give s a wider range in the intervalvalued optimal fractional removal levels of the p ollutants than GFWLAM, although at the cost of increasing uncertainty, in this p articular application. The current application of grey fuzzy multi-objective opt imization technique on waste load allocation demonstrates the modeling aspects o f uncertain membership functions for different management goals and shows the us efulness of solutions with a simplified hypothetical river system. Although the solutions obtained from the grey fuzzy multiobjective optimization model (i.e., multi-objective GFWLAM) provide more flexibility than those obtained from the gr ey fuzzy optimization model (i.e., GFWLAM), the application of multi-objective G FWLAM is limited to grey fuzzy goals expressed by linear imprecise membership fu nctions, whereas GFWLAM has the capability to solve the grey fuzzy optimization model with monotonic, nonlinear, imprecise membership functions with jl and jmn 1, in Eqs. (51) and (52). 6. CONCLUSION An overview of grey fuzzy optimization techniques are presented in this chapter. The concept of fuzzy decision is extended to grey fuzzy decision by considering the uncertainty in membership parameters using grey systems theory. A brief des cription of grey systems theory is presented as a prerequisite for understanding the grey fuzzy optimization technique. The grey fuzzy optimization model is fur ther enhanced to multi-objective framework to maximize the width of the optimal interval-valued decision variables providing latitude in decision making and to minimize the width of the goal fulfillment level for reducing the system uncerta inty. The concept of acceptability index for order relation between two partiall y or fully overlapping intervals is used to get a deterministic equivalent of th e grey fuzzy optimization model. Although the solutions obtained from the multiobjective optimization model provide more flexibility in decision making than th ose obtained by the grey fuzzy optimization model, the application of the multiobjective optimization model is limited to grey fuzzy goals expressed by linear imprecise membership functions, whereas the grey fuzzy optimization model has th e capability to solve the model with monotonic, nonlinear, imprecise membership functions also. The

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FUZZY MULTI-OBJECTIVE DECISIONMAKING MODELS AND APPROACHES Jie Lu1, Guangquan Zhang1, and Da Ruan2 Faculty of Information Technology, University of Technology, Sydney, Broadway, A ustralia 2Belgian Nuclear Research Centre (SCK•CEN), Belgium 1 Abstract: Multi-objective linear programming (MOLP) techniques are widely used to model ma ny organizational decision problems. Referring to the imprecision inherent in hu man judgments, uncertainty may be incorporated in some parameters of an establis hed MOLP model that is also called a fuzzy MOLP (FMOLP) problem. This chapter fi rst reviews the development of fuzzy multi-objective decision-making (FMODM) mod els and approaches and then proposes an effective way for an optimal solution in the FMOLP problem. By introducing an adjustable satisfactory degree , a new con cept of FMOLP and a solution transformation theorem are given in this chapter. T his chapter thus develops an interactive fuzzy goal multi-objective decision-mak ing method, which provides an interactive fashion with decision makers during th eir solution process and allows decision makers to give their fuzzy goals in any form of membership function. An illustrative example shows the details of the p roposed method. Fuzzy programming, multi-objective linear programming, interacti ve multiobjective decision-making method Key words: 1. INTRODUCTION Many organizational decision problems are involved in multiple objectives, calle d multi-objective decision making (MODM). Most MODM problems can be formulated b y multi-objective linear programming (MOLP) models. Referring to the imprecision and insufficient inherent in human judgments, uncertainty may be affected and C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 483

484 J. Lu et al. incorporated in some parameters of an MOLP model. Such a model is often called a fuzzy MOLP (FMOLP) model. Various methods have been proposed from the literatur e to derive a satisfaction solution of an MOLP problem for decision makers based on their subjective value judgment and preference. Two main types of such metho ds are goal programming and interactive programming (Hwang and Masud, 1979). In general, there is no unique solution for both MOLP and FMOLP problems. To obtain a satisfactory solution of an FMOLP problem for a particular decision maker inv olves a lot of interaction to carry out the decision maker’s preference for a solu tion. When both the parameters in the model and the goals given by a decision ma ker are with uncertainty the interactive solution procedure may become very comp lex, and therefore, more efficient FMOLP methods are needed. Many optimization m ethods and techniques for modeling and solving FMOLP problems have been proposed (Carlsson and Fuller, 1996; Inuiguchi and Ramik, 2000; Lai and Hwang, 1994; Sak awa, 1993a). Fuzzy numbers seem promising to model and solve an FMOLP problem. M any applications have also proved it applicable for dealing with human decisionmaking problems in most practical situations (Bellmann and Zadeh, 1970; Sakawa, 1993b). Tanaka and Asai (1984) formulated FMOLP problems by using triangular fuz zy numbers to describe the fuzzy parameters in both objective functions and cons traints. Lai and Hwang (1992) also modeled FMOLP problems by using triangular fu zzy numbers and solved FMOLP problems by the fuzzy ranking concept as well to ha ndle imprecise constraints. Luhandjula (1987) proposed the concepts of -possible feasibility and -possible efficiency based on fuzzy numbers and used the two co ncepts to solve the FMOLP problem by transferring it into an auxiliary crisp MOL P problem. Furthermore, Slowinski (1990) proposed an interactive method (FLIP) f or solving MOLP problems with fuzzy parameters in the objective functions and on the both sides of the constraints. Rommelfanger (1989, 1990) presented a method (FULPAL) for solving (multi-criteria) linear programs, where the right-hand sid es as well as the parameters in the constraints and/or the objective functions m ay be fuzzy. Similarly, Ramik and Rommelfanger (1993, 1996) proposed a unified a pproach based on the fuzzy inequality relations for the fuzzy mathematical progr amming problem in which fuzzy parameters may have nonlinear membership functions . In particular, Inuiguchi and Ramik (2000) reviewed some fuzzy linear programmi ng methods and techniques from a practical point of view and introduced the gene ral history and the approaches of fuzzy mathematical programming. In the meantim e, goal programming (Charnes and Cooper, 1977) as an effective method has been

Fuzzy MODM Models and Approaches 485 successfully applied in solving FMOLP problems. Kuwano (1996) applied the concep ts of the -optimal solution and the restricted -optimal value at the -optimal so lution to establish a goal programming approach for solving FMOLP problems. Saka wa and Nishizaki (2000) pushed the work forward based on their previous results (Sakawa, 1993a; Sakawa and Yano, 1990) by defining two new concepts for FMOLP ba sed on fuzzy goals. One is defined by maximizing the minimal fuzzy goal and the other by maximizing the sum of fuzzy goals. They then developed two computationa l methods for obtaining the solutions for FMOLP problems. More importantly, Rami k (2000) generated a standard goal programming problem with alternatives and goa ls being fuzzy sets, and the satisfaction of a goal by a fuzzy objective functio n is also expressed by a fuzzy relation; he proposed a unifying approach coverin g several approaches known from the literature. Although these methods are effic ient to solve FMOLP problems, there are two limitations in their current results . One is that only some specialized forms of membership functions such as a tria ngular form were used to deal with fuzzy parameters and fuzzy goals. This may re strict the use of other forms of membership functions to describe the parameters in modeling an FMOLP problem and to express their goals by decision makers in s olving the FMOLP problem. The second limitation is that the values of objective functions, in corresponding to a satisfactory solution of an FLOMP problem, are only described by some crisp values, which is sometimes not appropriate in pract ice. Since a decision problem is formulated with uncertainty and its solution is received with fuzzy values, it is more reasonable to provide the values of the objective functions with a range in values. This study, therefore, develops a ge neralized fuzzy goal fuzzy multiobjective optimization method to assist decision makers to obtain satisfactory solutions for an FMOLP problem. The method can so lve the FMOLP problem with whatever the parameters of both objectives and constr aints are described in any form of membership functions. The method also allows decision makers to provide their fuzzy goals for the objectives of their decisio n problems by linguistic terms by any form of membership functions. By introduci ng an adjustable satisfactory degree , the obtained values of objective function s, corresponding to a solution, can be described by fuzzy values in which a real number is as a special case. Moreover, the generalized fuzzy goal fuzzy multi-o bjective decisionmaking method has the features of interaction with decision mak ers during a solution process.

486 J. Lu et al. The results reported here are our continuing research, and a summary about our p revious reports is in (Lu et al., 2007; 2006; Wu et al., 2003; 2004a; 2004b; 200 6; Zhang et al., 2002; 2003). This chapter first gives a general FMOLP model whe re fuzzy parameters of objective functions and constraints are described by memb ership functions. To solve such an FMOLP problem, an optimal solution concept, a general solution transformation theorem, and a related workable solution transf ormation theorem are then developed. Based on these theories, an FMOLP problem c an be transformed into an MOLP problem. Therefore, an optimal solution of an FMO LP can be obtained through solving an associated MOLP problem. Under this princi ple, an interactive FMOLP method is presented by 11 steps within two stages. Fin ally a numeral example illustrates the proposed FMOLP method. 2. FUZZY MULTI-OBJECTIVE DECISIONMAKING MODEL This section introduces a set of fuzzy multi-objective linear programming models . It then gives the concepts of optimal solutions for such kinds of problems. Th ese models will be applied in the following sections to develop related methods and algorithms to achieve an optimal solution for an FMOLP problem. 2.1 Model and Pareto Optimal Solution for General FMOLP Problems Consider the following fuzzy multi-objective linear programming (FMOLP) problem: Maximize (FMOLP) subject to ~,x c n F i 1 ~1i x i , c 0, n i 1 ~2 i x i , c n , i 1 ~ki x i c T (1) ~ Ax ~ b,x where

Fuzzy MODM Models and Approaches 487 ~ c ~ b ~ c11 ~ c 21 ~ c12 ~ c 22 ~ ~ ck 1 ck 2 ~ ~ ( b1 , b2 , ~ c1n ~ ~ c2 n , A ~ c kn ~ a 11 ~ a 21 ~ a 12 ~ a 22 ~ ~ a m1 a m 2 ~ a 1n ~ a2n , ~ a mn ~ , b m )T F ( R m ), ~ and c sj , aij F * ( R), s 1, 2, , k , i 1, 2, ~ , m, j 1, 2, ~ ~ , n. For the sake of simplicity, we set X x ; A x b , x 0 and assume that ~ ~ X is co mpact. In an FMOLP problem, for each x X , the value of the ~ objective function c , x F is a fuzzy number. Thus, we introduce the following concepts of optimal solutions to FMOLP problems. DEFINITION 1. A point x* R n is said to be a compl ete optimal solution to the FMOLP ~ ~ ,x problem if it holds that ~ , x * c c F for all x X . F DEFINITION 2. A point x* R n is said to be a Pareto optimal solution to the FMOL P ~ ~ , x* problem if there is no x X such that ~ , x F c c holds. F DEFINITION 3. A point x* R n is said to be a weak Pareto optimal solution to the ~ ~ ~ * c ,x holds. FMOLP problem if there is no x X such that c , x F F The ba sic ideas to solve the FMOLP problem are (1) to transform it into an associative crisp MOLP problem. (2) As MOLP problems have been well studied, a Pareto optim al solution of the MOLP problem can be obtained. (3) Through setting up the rela tionship between the solution of the FMOLP and the solution of the associative M OLP, the original FMOLP problem can be solved. Therefore, we first consider the following MOLP problem that is associated with the FMOLP problem shown in (1):

Maximize (MOLP) subject to A x b , A x b , x 0 , L L R R c L ,x , c R ,x T (2) [ 0 ,1 ] where

488 c 11 CL L c 21 L ck1 L J. Lu et al. c 12 L c 22 L ck 2 L c 1n L c 2n L c kn L c 11 R c 12 c 22 R ck 2 R R c 1n c2n R c kn R R , CR c 21 R ck1 R a 11 A L L a 21 L a m1 L a 12 L a 22 L a m2 T L a 1n L a 2n L a mn L a 11 R a 12 a 22 a m2 ,b m R T R

a 1n a 2n a mn R R R , AR a 21 a m1 R R R R b L b 1 ,b 2 , L L ,b m L , b R b 1 ,b 2 , R R In the following, we introduce the concepts of optimal solutions of the MOLP pro blem. DEFINITION 4. A point x * L R n is said to be a complete optimal solution to the MOLP problem if it holds that ( c L , x * , c R , x * )T ( c L , x , c R , x )T , for all x X x; A x b ,A x L R b ,x R 0, [0, 1] and

[0, 1]. DEFINITION 5. A point x * R n is said to be a Pareto optimal solution to the MOL P problem if there is no x X such that L * R * T L R T ( c ,x , c ,x ) ( c ,x , c , x ) , 0 , 1 holds. DEFINITION 6. A point x * R n is said to be a weak Pareto optimal solution to the MOLP problem if there is no x X such that L * R * T L R T ( c ,x , c ,x ) ( c ,x , c , x ) , 0 , 1 holds. THEOREM 7. Let x* R n be a fe asible solution to the FMOLP problem. Then 1. x * is a complete optimal solution to the FMOLP problem, if and only if x * is a complete optimal solution to the MOLP problem.

Fuzzy MODM Models and Approaches 489 2. x * is a Pareto optimal solution to the FMOLP problem, if and only if x * is a Pareto optimal solution to the MOLP problem. 3. x * is a weak Pareto optimal s olution to the FMOLP problem, if and only if x * is a weak Pareto optimal soluti on to the MOLP problem. Proof. The proof follows directly from Definitions 1 6. 2.2 Model and Pareto Optimal Solution for FMOLP Problems [0, Obviously, a feasible solution must satisfy the constraints for all 1]. Howe ver, this is a too strong condition to get an optimal solution. We ~ therefore c onsider a typical parameter ci represented by a fuzzy number ci . L R The possib ility of such a parameter ci taking values in the range [ ci , ci ] is or above. While the possibility of ci taking values beyond [ ci L , ci R ] is less than . Thus, one would be generally more interested in a solution > 0. As a special us ing parameters ci taking values in [ ci L , ci R ] with case, if the parameters involved are either a real number or a fuzzy number with a triangular membership function, then, we will have the usual nonfuzzy optimization problem (suppose w e choose = 1). To formulate this idea, we introduce the following FMOLP model. Maximize (FMOLP ) subject to ~ Ax ~,x c n F i 1 ~i x i c 0 0, 1 (3) ~ b,x where ~ c11 ~ c 21 ~ c12 ~ c 22 ~ c1n ~ ~ c2 n , A ~ c kn ~ c ~ b ~ a 11 ~ a 21 ~ a 12 ~ a 22 ~ ~ ck 1 ck 2 ~ ~ ( b1 , b2 , ~ ~ a m1 a m 2 ~ a 1n ~ a2n , ~ a mn ~ , b m )T F * ( R m ), 1, 2 , ,k , i 1, 2 , , m, j 1, 2 , , n. and ~sj , a ij c ~ F * ( R ), s

490 J. Lu et al. Now, associated with the FMOLP problem, consider the following MOLP problem, maximize (MOLP ) subject to A x L L c L ,x , c R ,x b ,A x R T (4) 0, [ ,1 ] b ,x R where L c 11 L c 12 L c 22 L ck2 L c 1n L c 2n L c kn c 11 R c 12 R c 22 R ck2 R c 1n R c 2n R c kn R CL L c 21 L ck1 , CR R c 21 R ck1 L a 11 L a 12 L a 22 L a 1n L a 2n a 11 R a 12 a 22 R a m2 R R

a 1n a 2n R a mn R R AL L a 21 , AR a 21 R a m1 R a m1 L a m2 L L a mn L b L b 1 ,b 2 , L ,b m L T , b R b 1 ,b 2 , R R ,b m R T

. In the following, we introduce the concepts of optimal solutions of the MOLP pro blem. DEFINITION 8. A point x * R n is said to be a complete optimal solution to the ~ c ,x F F ~ FMOLP problem if it holds that c , x * for all x ~ X . DEFINITION 9. A point x * R n is said to be a Pareto optimal solution to the FMOLP ~ ~ ~ * holds. problem if there is no x X such that c , x F c ,x F DEFINITION 10. A point x * R n is said to be a weak Pareto optimal solution to t he FMOLP problem if there is no x ~ X ~ such that c , x F ~ * c ,x F holds.

Fuzzy MODM Models and Approaches 491 DEFINITION 11. A point x * R n is said to be a complete optimal solution to the MOLP problem if it holds that ( c L , x * , c R , x * )T ( c L , x , c R , x )T , for all x X x; A x L L R R b ,A x b ,x 0, [ , 1 ] and [ , 1]. DEFINITION 12. A point x * R n is said to be a Pareto optimal solution to the MO LP problem if there is such that no x X L * R * T L R T ( c ,x , c ,x ) ( c ,x , c , x ) , , 1 holds. DEFINITION 13. A point x* R n is said to be a weak Pareto optimal solution to the MOLP problem if there is no x X such that L * R * T L R T ( c ,x , c ,x ) ( c ,x , c , x ) , , 1 holds. THEOREM 14. Let x * R n be a feasible solution to the FMOLP problem. Then 1. x * is a complete optimal solution to the FMOLP problem, if and only if x * is a co mplete optimal solution to the MOLP problem. 2. x * is a Pareto optimal solution to the FMOLP problem, if and only if x * is a Pareto optimal solution to the MO LP problem. 3. x * is a weak Pareto optimal solution to the FMOLP problem, if an d only if x * is a weak Pareto optimal solution to the MOLP problem. Proof. The proof follows directly from Definitions 8 13 and Theorem 14. In this section, we have addressed the FMOLP problem and have introduced the concepts of complete o ptimal solution, Pareto optimal solution, and weak Pareto optimal solution for F MOLP, FMOLP , MOLP, and MOLP . We have also proposed an efficient approach for s olving the FMOLP and FMOLP problems, which is to transform them into the associa tive crisp MOLP and MOLP .

492 J. Lu et al. 3. SOLUTION TRANSFORMATION THEORIES FOR FUZZY MULTIOBJECTIVE DECISION-MAKING PROBLE MS As outlined in Section 2, the possible values of parameters in the FMOLP are app ropriate to be represented by fuzzy numbers. Here we will show how a fuzzy numbe r parameters-based FMOLP problem is transformed into an associated MOLP problem. 3.1 General MOLP Transformation Theories Consider the situation in which all parameters of the objective functions and th e constraints are fuzzy numbers represented in any form of membership functions. Such FMOLP problems can be formulated as follows: ~ Lemma 15. If a fuzzy set c on R has a trapezoidal membership function (see Figure 1): 0 x cL cL cL x cR cR cR x x x cL cL cR cL ~( c cL x) x cR x c 0 R c R 1 cL cL cR cR Figure 1. Trapezoidal membership function

Fuzzy MODM Models and Approaches 493 and there is x * c ,x R X n such that c L , x , (0 1 ) , and c ,x c ,x R L * , c ,x c ,x R * L c ,x , L * , c ,x R * for any x X n , then c ,x R L c ,x R L * c ,x c ,x *

for any [ , ]. Proof. c As a -section of the trapezoidal fuzzy set ~ is c L c L c L c L and c R c R c R c R Therefore, we have c ,x L c L c L L ,x L c ,x L L c ,x cL , x c L , x* L L

c ,x c ,x cL , x L c1 , x* c c ,x * c ,x L * c ,x L * from c1 , x L c1 , x R L * * , c ,x L c ,x L * and 0 1, we can prove c R , x c ,x from a similar reason.

494 J. Lu et al. THEOREM 16. ~ If all the fuzzy parameters ~sj , aij , and bi have trapezoidal me mbership c ~ functions: 0 t z L t z z R L z L t t t R z z z t L z ~(t z L z L ) t z L R (5) z R R z 0 R z R z ~ ~ ~ where ~ denotes c sj , a ij or bi respectively, then the space of feasible z solutions X is defined by the set of x X with xi, for i = 1,2,…,n satisfying n j 1 n j 1 n j 1 n j 1

a ij L x j a ij R x j a ij L x j a ij R x j 0 bi L bi R bi L bi R . (6) xi

Fuzzy MODM Models and Approaches 495 Proof. From Theorem 7, X is defined by n n X {x Rn j 1 aijL x j bi L , j 1 aijR x j bi R , x 0 (7) [ , ] and i 1, 2, , m}. That is, X is the set of x Ii n Rn with x 0 and satisfying bi L 0, J i n a ij L x j a ij R x j bi R 0 j 1 j 1 (8) [ , ] and i 1,2, ,m. For the fuzzy sets with trapezoidal membership functions, we have a ij L a ij L a ij L ( ) a ij L (9) a ij R a ij R a ij R ( ) a ij R ,

bi L bi R bi L bi R bi L bi R ( ) bi L (10) ( ) bi R Substituting Eqs. (9) and (10) into (8), we have Ii n [ a ij L a ij L ( ) a ij L ] x j n [ bi L bi L ( ) bi L ] j 1 n (11) a ij L x j bi L a ij L x j bi L j 1 j 1

496 n J. Lu et al. a ij R a ij R bi R bi R Ji [ ( ) a ij R ] x j n [ ( ) bi R ] j 1 n (12) a ij R x j bi R a ij R x j bi R j 1 j 1 Now, our problem becomes to show that [ , ] and i 1, 2, n Ii 0, J i 0, , m if (6) is satisfied. From Eq. (6), we have aij L x j j 1 bi L 0 (13a) n aij R x j bi R j 1 0 (13b) n aij L x j bi L

j 1 0 (13c) n aij R x j bi R j 1 0. (13d) Thus, from Eqs. (13a) and (13c), we have for any i = 1,2,…,m Ii n j 1 [ , ] and a ij L x j bi L n j 1 a ij L x j bi L 0 and from Eqs. (13b) and (13d), we have for any Ji n j 1 [ , ] and i = 1,2,…,m n a ij R x j bi R a ij R x j bi R 0. j 1 Corollary 17. ~ c ~ If all the fuzzy parameters ~sj , aij and bi have piece-wise trapezoidal membership functions

Fuzzy MODM Models and Approaches 497 t z t L 0 0 1 L z 1 2 0 L z 0 1 t z L0 0 z L0 z L1 z L2 ~(t z z L1 t z L1 1 z L 1 t z L 2 ) n n 1 1 z Ln t z R n 1 t t z Rn z R n 1 (14) z Rn z Rn 1 n 1

z R n 0 z R 1 z R 0 t z R 0 0 z R 1 R 0 t z t R 0 0 ~ z where ~ denotes ~sj , a ij or bi respectively, then the space of feasible z c ~ solutions X is defined by the set of x X with xi, for i = 1,2,…,n satisfying n j 1 n j 1 n j 1 n j 1 n j 1 n j 1 a ij L0 x j a ij R0 x j a ij L1 x j a ij R1 x j bi L0 bi R0 bi L1 bi R1 bi L n bi Rn 0 . (15) a ij L n x j a ij Rn x j xi

498 J. Lu et al. THEOREM 18. Let all the fuzzy parameters be piece-wise trapezoidal membership fu nctions in FMOLP : 0 1 0 t t z L 0 z L0 t z L1 z L1 2 z L0 1 0 z L0 z L2 ~(t z z L1 t z L1 1 z L1 z Ln t z L2 z Rn z Rn 1 ) 1 n n 1 1 t t (16) z Rn z Rn 1 t z Rn

1 n 1 z Rn 0 z R1 0 z R0 t z R0 0 z R 1 t z t R 0 z R0 If a point x * R n be a feasible solution to the FMOLP problem, then x is a comp lete optimal solution to the problem if and only if x* is a complete optimal sol ution to the MOLP problem: *

Fuzzy MODM Models and Approaches Maximize c c c c c c subject to (MOLP ) n j 1 n j 1 L 0 R 0 L 1 R n 1 L n R n 499 ,x ,x ,x ,x ,x ,x bi L0 a ij L 0 x a ij R 0 j (17) x j bi R 0 n j 1 n j 1 n j 1 n j 1 a ij L 1 x a ij R 1 x a ij L n x a ij R n x 0 j bi L1 b i R1 bi L n bi Rn j j j xi where 0 1 n 1 n 1. Proof. If x* is an optimal solution to the FMOLP problem, then for any ~ ~ , x . Therefore, for any x X , we have ~ , x * [ , 1], c F c F ( n i 1 ~ x* ) L ci i

( n i 1 ~ x )L ci i and ( n i 1 ~ x* ) R ci i ( n i 1 ~ x )R ci i

500 J. Lu et al. that is n i 1 ci L x* i n i 1 ci L xi and n i 1 ci R x* i n i 1 ci R xi Hence x* is a complete optimal solution to the MOLP problem by Definition 11. If x* is a complete optimal solution to the MOLP problem, then for all x X , we ha ve ( c L i ,x * , c R i ,x * ) T ( c L i ,x , c R i ,x ) , i T 0 , 1, ,n

that is n n n n j 1 c j Li x j * j 1 c j Li x j , j 1 c j Ri x j 1, 2 , * j 1 c j Ri x j , i 1, 2 , , n For any [ , 1], there exist i , n so that i 1 , i . c As ~ has a piece-wise trapezoidal membership function, we have cL i i 1 i 1 c L1 c Li 1 c Li 1 and cR i i 1 i 1

c Ri c Ri 1 c Ri 1 From Lemma 15, we have n i 1 ci L x* i n i 1 ci L xi and n i 1 ci R x* i n i 1 ci R xi ,

Fuzzy MODM Models and Approaches 501 for any problem. [ , 1]. Therefore, x* is an optimal solution to the FMOLP THEOREM 19. Let all the fuzzy parameters be piece-wise trapezoidal membership fu nctions in FMOLP : 0 1 L z 1 2 L z 2 ~(t z 0 L z 0 1 L z 1 t t z L 0 z t L 0 0 z L 0 z L 1 t z L 1 1 z L 1 t z L 2 ) 1 n n 1 z Ln t z Rn z R n t t z Rn . z R n 1

(18) z Rn 1 0 R z 1 z Rn 1 R z 0 1 n 1 t z R 0 0 z R 1 t z R 0 R 0 0 ~ z t Let a point x* X be any feasible solution to the FMOLP problem. Then x* is a Par eto optimal solution to the problem if and only if x* is a Pareto optimal soluti on to the MOLP problem:

502 J. Lu et al. Maximize c L0 , x R ,x 0 L c 1 ,x c c Rn c 1 ,x subject to (MOLP ) L ,x n R c n ,x n a ij L0 x j j 1 n a ij R0 x j j 1 n a ij L1 x j j 1 n j 1 n j 1 n j 1 bi L0 bi R0 bi L1 bi R1 bi L n bi Rn a ij R1 x j a ij L n x j a ij Rn x j 0 xi (19) where 0 1 n 1 n 1. Proof. ~ Let x * X be a Pareto optimal solution to the FMOLP problem. On the con trary, we suppose that there exists an x X such that ( cLi , x * , cRi , x * )T Therefore ( c Li , x , cRi , x )T , i 0, 1, , n (20) 0 ( cLi , x cLi , x* , cRi , x

cRi , x* ,)T , i 0, 1, 2, , n (21)

Fuzzy MODM Models and Approaches 503 Hence 0 cLi , x cLi , x* , 0 cRi , x cRi , x* , i 0, 1, 2, , n (22) That is c Li , x c Li , x * , c Ri , x c R1 , x * , i 1, 2 , ,n By using Lemma 15, for any [ , 1], we have c ,x L * c ,x L and c ,x R * c ,x R ~ ~ * . However, this contradicts x X is a Pareto optimal solution mal solution to the MOLP problem. problem, then there exists an ~ ~ , x F [ , 1], we have F n i 1 ( ~ x* ) L ci i ( the assumption that c ,x that to the FMOLP problem. Let x X If x* is not a Pareto optimal ~ * . Therefore, for any c ,x is c , x F F ~ * be a Pareto opti solution to the x X such that c

n i 1 ~ ci xi ) L and ( n i 1 ~ x* ) R ci i ( n i 1 ~ ci xi ) R That is c ,x L * c ,x 0 L and 1 c ,x R * c ,x n R Hence, for n 1 1, we have 0, 1, , n ( c Li , x * , c Ri , x * )T ( c Li , x , c Ri , x )T , i which contradicts the assumption that x* to the MOLP problem. X is a Pareto optimal solution THEOREM 20. Let all the fuzzy parameters be piece-wise trapezoidal membership fu nctions in FMOLP :

504 0 1 L z 1 2 L z 2 0 L z 0 1 L z 1 J. Lu et al. t t z L 0 z t L 0 0 z L 0 z L 1 t z L 1 1 z L 1 t z L 2 (23) L z n ~(t z ) 1 n R z n 1 n 1 R z n t t R z n R n 1 . t z

R n 1 n 1 z R n z 0 R z 1 1 R z 0 t z R 0 0 z R 1 R 0 t z t R 0 0 z and a point x* X be a feasible solution to the FMOLP problem. Then x* is a weak Pareto optimal solution to the problem if and only if x* is a weak Pareto optima l solution to the MOLP problem: Maximize c c c c c n c R 0 L 1 R L 0 ,x ,x ,x ,x n 1 L n R n ,x ,x L 0 R 0 L 1 subject to (MOLP ) j 1 n a ij a ij j 1 n

x x x j bi L 0 bi R0 bi L 1 bi R1 bi L n bi R n (24) j a ij j 1 n j a ij j 1 n R 1 L n R n x x x j a ij j 1 n j a ij j 1 xi j 0

Fuzzy MODM Models and Approaches 505 Proof. See Theorem 19. Therefore, if we use existing methods to get a complete o ptimal solution x * to the MOLP problem, then x * is a complete optimal solution to the FMOLP problem. This gives a way to solve the FMOLP problems, which will be used in developing detailed FMOLP algorithms and methods. 4. FUZZY-GOAL MULTI-OBJECTIVE DECISIONMAKING MODEL Decision makers may want to specify their fuzzy goals for the objective function s in dealing with the FMOLP problem (21) under some circumstances. The key idea behind goal programming is to get the optimal solution that has the minimized de viations from goals set by decision makers. In standard goal programming, goals need to be given by precise data. In practice, it is often difficult for a decis ion maker to provide a precise attainment for each objective function. Applying fuzzy set theory into goal programming makes it possible for decision makers to indicate their vague aspirations, which can be qualified by linguistic terms. Su ch goals can be expressed as, for instance, “possibly greater than g 1 ,” “around g 2 ” or “substantially less than g 3 .” These types of linguistic terms can then be quali fied by eliciting membership functions of fuzzy sets. Considering the FMOLP prob lem for the fuzzy multiple objective ~ , x , any decision maker can specify fuzz y goals c functions F ~ g , g ,… , g T under a satisfactory degree ~ ~ ~ that refl ects the desired g 1 2 k values of the objective functions of the decision maker . These fuzzy goals can be represented by fuzzy numbers with any form of members hip ~ ~ ~ functions. By defining a fuzzy deviation function D c , x F , g as a f uzzy ~ difference between the fuzzy objective function c , x F and fuzzy ~ ~ ~ ~ goal s g g 1 , g 2 ,… , g k T , the FMOGP problem under a satisfactory degree is formul ated as follows:

506 J. Lu et al. Minimize D c, x F , g (FMOGP ) subject to Ax b (25) x that is, find an x* ~ X 0 ~ D ~, x c ~ ,g , which minimizes x* arg min x X F or (26) ~ D ~, x c F ~ ,g . ~ ,g Normally, the fuzzy distance function of deviations of individual goals, ~ ~ D c,x ~ ,g F ~ D ~, x c F is defined as a maximum i 1,….k ~ max Di n j 1 ~ ~ cij x j , gi . (27) By Eq. (26), the FMOGP problem (25) is converted as follows: Min max ~ Di n i 1 ,… ,k

~ij x j , g i ~ c j 1 ~ ~ subject to A x b x 0 (28) where n n n Di j 1 cij x j , g i max 0,1 j 1 cij x j L g iL , j 1 cij x j R g iR (29) max ciL x g iL , ciR x g iR 0,1 i 1,… , k . From Eq. (29), the optimal solution of Eq. (28) can be obtained by solving the f ollowing GP model:

Fuzzy MODM Models and Approaches 507 min max c iL x i 1,…,k ,1 g iL , c iR x g iR (GP -1 ) (30) subject to A L x x 0 bL , A R x bR , ,1 or min max g iL i 1,…,k ,1 c iL x, g iR c iR x (GP -2 ) (31) subject to A L x x 0 bL , A R x bR , ,1 where L c1 L c2 L ck L c 11 L c 21 L c 12 L c 22 L c 1n L c 2n c1 R , c2 ck R R c 11 R c 21 R c 12 R c 22 R c 1n R c 2n R ck1 L

ck2 L c kn L ck1 R ck2 R c kn R The adoption of GP -1 (30) or GP -2 (31) for solving the FMOGP ~ ~ ~ problem dep ends on the relationship of c , x F and g ; i.e., if ~ , x F g , then c GP (30) is used; otherwise, GP -2 (31) is adopted. Hence, when we get a complete optimal solution x * to the goal programming problem, x * is a complete optimal solutio n to the FMOLP problem. -1

508 J. Lu et al. 5. AN INTERACTIVE FUZZY-GOAL FUZZY MULTI-OBJECTIVE DECISION-MAKING METHOD Fuzzy Goa l-Based Interaction 5.1 Many decision makers prefer an interactive approach to finding an optimal soluti on for their decision problem as such an approach enables them to directly engag e in the problem-solving process. This section proposes an interactive algorithm based on the fuzzy goal approximation algorithm. This algorithm not only allows decision makers to give their fuzzy goals but also allows them to continuously revise and adjust their fuzzy goals. Decision makers can then explore various op timal solutions under their goals and choose the most satisfactory one. From the definitions of both FMOLP and MOLP problems, any ~ ~ ~ ~ decision maker can set up their fuzzy goals g g 1 , g 2 ,… , g k T under a satisfactory degree . Its cor responded optimal solution, which results in the objective values being the near est to the fuzzy goals, is obtained by solving the following minimax problem: min max MOLP subject to x X x R A x n L C x R C x L g R , g L ,1 b ,A x L R (32) b ,x R 0, ,1 where g L g 1 ,g 2 , L c 11 L c 12 L c 22 L L ,g k L T

, gR R R g 1 ,g 2 , R ,g k T L c 1n L c2n R c 11 R c 12 R c 1n CL L c 21 , CR c 21 R ck1 R c 22 R ck 2 R c2n R c kn R ck1 L ck 2 L c kn L

Fuzzy MODM Models and Approaches L a 11 L a 12 L a 22 L a 1n L a 2n 509 R a 11 R a 12 R a 22 R a m2 R a 1n R a 2n R a mn A L L a 21 , AR R a 21 R a m1 (33) a m1 L a m2 L ,b m L a mn T L bL L L b 1 ,b 2 , , bR R R b 1 ,b 2 , R ,b m T . Let the interval [ , 1] be decomposed into l mean sub-intervals with (l+1) nodes i i 0 , ,l that are arranged in the order of 0 1 l 1. Based on the current decompositions, we denote: ciL j x g iL ji ciR j x g iR ji x Xl (MOLP m) l min max subject to

, i 1,2, ,k, j 1, 2, , l, (34) where X l l i X i ,X i x R n A Li x

b Li , A Ri x b Ri , x 0 , ,1 . 5.2 Description of the Algorithm This algorithm consists of 11 steps within two stages. Stage 1 aims to find an i nitial optimal solution for the problem. Stage 2 is an interactive process in wh ich when a decision maker specifies a set of fuzzy goals for related objective f unctions, an optimal solution is generated. By revising fuzzy goals, this algori thm will provide the decision maker with a series of optimal solutions from whic h the decision maker can select the most suitable one on the basis of preference , judgment, and experience. The algorithm is described as follows:

510 J. Lu et al. Stage 1: Initialization Step 1. Select an initial satisfactory degree 0 1 , give the ~ b ~ membership function of c for ~ x f ~ ~ ~ ~ c x , a and b for a x , and set weights for objective functions by the decision maker. Step 2. Set l 1 , then so lve ciL j x ciR j x x l (MOLP ) l max , Xl i 1, ,k; j 0,1, , l, (35) subject to. with the solution x l , where x is subject to the constraint x x 1 , x 2 ,… , x n l l , and the solution obtained X . x 2l Step 3. Solve (MOLP )2l with the solution 2l , subject to the constraint x X . The interval [ , 1] is further split. Suppose there are l 1 nod es i i 0, 2, 4,… , 2l in the interval [ , 1], and l new nodes i 1, 3, … , 2l 1 are i nserted. The relationship between the new i nodes and previous ones is: 2i 2i 1 2i 2

2 , i 0, 1, … , l 1 . (36) Therefore, each of the fuzzy objective functions is converted into 2 2l 1 non-fu zzy objective functions, and the same conversion ~ ~ happens for the constraints ai x bi . The solution x 2 l is now based on the set of updated (including orig inal) nonfuzzy objective functions and nonfuzzy constraints. , then x 2 l is the final solution of the MOLP Step 4. If x 2 l x l problem. Otherwise, update l to 2l and go back to Step 3. Step 5. If the corresponded Pareto optimal solution x * exists, go forward to Step 6. Otherwise, the decision maker must go back to S tep 1 to reassign a degree (give a higher value for the degree ). Step 6. If the decision maker is satisfied with the Pareto optimal solution, the interactive p rocess terminates. Otherwise, go to Stage 2.

Fuzzy MODM Models and Approaches 511 Stage 2: Iteration As the decision maker is not satisfied with the obtained opti mal solution in the Initialization stage (or the previous iteration phase), the decision maker specifies fuzzy goals (or revised current goals) for the fuzzy ob jective functions. A new compromise solution is then generated. This process wil l terminate when the decision maker finds a satisfactory solution. Step 7. Give a set of new fuzzy goals or revise current fuzzy goals according to the decision maker. At the same time, a satisfactory degree can be revised as well. The orig inal decision problem is therefore covered into an (MOLP m) l problem. Step 8. S et l 1 ; solve (MOLP m) l with the solution x l , subject to the constraint x X l . Let and 1 1 in the interval [ , 1]; each fuzzy objective function ~ ~ ~ ~ ~ g 1 , g 2 ,… , g k T , and related constraints ci x under the fuzzy goal g are con verted into non-fuzzy forms. 0 ~ fi x Step 9. Solve (MOLP m) 2l with the solution x 2 l subject to the constraint x X 2 l . Similar to Step 6, the interval [ , 1] is further split, and new nodes are inserted further. Fuzzy objective functions under related fuzzy goals and const raints are converted into non fuzzy again. A new solution x 2 l is generated. , then x 2 l is the final solution of the MOLP m Step 10. If x 2 l x l problem. Ot herwise, update l to 2l and go back to Step 9. Step 11. If the decision maker is satisfied with the current Pareto optimal solution obtained in Step 10, the int eractive process terminates, and the current optimal solution is the final satis factory solution to the decision maker. Otherwise, go back to Step 7. We now giv e another explanation for this algorithm: Definition 1 is about ranking two n-di mensional fuzzy numbers under a satisfactory degree . This definition is the fou ndation for the comparison of fuzzy objective functions and the left- and righthand sides of fuzzy constraints in an FMOLP problem. In Step 5 of this method, i f the Pareto optimal solution does not exist under a satisfactory degree , repla cing this with a higher value may derive a Pareto optimal solution.

512 J. Lu et al. In Step 7 of the algorithm, the decision maker can improve goals for some unsati sfactory objectives by sacrificing the goals of others. The new fuzzy goals can be given directly by a new fuzzy number vector or by increasing/decreasing the v alues of its corresponded objective functions in a current Pareto optimal soluti on. Figure 2 shows the flowchart of the fuzzy goal interactive algorithm. 6. A NUMERAL EXAMPLE To illustrate the interactive fuzzy-goal multi-objective algorithm, we consider the following FMOLP problem with two fuzzy objective functions and four fuzzy co nstraints: ~ f max ~1 x f2 x ~ ~ 4 x1 2 x 2 ~ ~ - 2 x1 4 x 2 ~ max f x ~ max c 11 x 1 ~ c 21 x 1 ~ c 12 x 2 ~ c 22 x 2 max ~ a 11 x 1 ~ a 21 x 1 ~ a 12 x 2 ~ a 22 x 2 ~ a 32 x 2 ~ a 42 x 2 0 ~ - 1 x1 ~ 1 x1 ~ 4 x1 ~ 3 x1 ~ ~ 3 x 2 b1 ~ ~ 3 x 2 b2 ~ ~ 3 x 2 b3 ~ ~ 1 x 2 b4 ~ 21 ~ 27 45 30 ~ ~ subject to ~ a 31 x 1 ~ a 41 x 1 x1 0; x 2 The membership functions of the parameters of the objective functions and constr aints are set up as follows:

Fuzzy MODM Models and Approaches Start 513 Set up the FMOLP model, i.e., input the ~ ~ , membership functions of c for ~ x f cx ~ ~ ~ a and b for ~ ax b Set weights for each ~i x f ~x ci Specify an initial value of the degree 0 1 ~ * f x Calculate the max fuzzy objective functions ~ max Cx of the FMOLP problem under ~ the constraints a x ~ b N Solution exists? Y Y Satisfy solution? N ~ ~ ~ ~ Specify new fuzzy goals g g1, g 2 , , g k T for objective functions base d on the current fuzzy Pareto optimal solution Calculate the fuzzy Pareto optimal solution based on the current fuzzy goals of objective functions and degree specified above N Satisfy solution? Y The interactive process stops here and the final solution is shown End Figure 2. Flow chart for the fuzzy goal interactive algorithm

514 0 x2 1 36 9 7 x 2 20 x 3 or 6 3 x 4 x 4 4 x 6 x ~ c12 J. Lu et al. 0 x2 1 3 1 16 x 2 12 x 1 or 4 1 x 2 x 2 2 x 4 x ~ c11 x x ~ c21 x 0 6.25 x 2 2.25 1 x2 1 3 x x 2.5 2.5 or - 1 x x 2 2 2 x 1 ~ c22 x 0 x2 9 7 1 36 x 2 20 x 3 or 6 3 x 4 x 4 4 x 6 x ~ a11 x 0 4 x2 / 3 1 x 2 0.25 0.75 2 or 0.5 x 1 x 1 1 x 0.5 2 x x ~ a12 x 0 x2 4 / 5 1 25 x 2 16 x 2 or 5 2 x 3 x 3 3 x 5 x ~ a 21

x 0 x 2 0.25 / 0.75 1 4 x2 3 x 0.5 or 2 0. 5 x 1 x 1 1 x 2 x ~ a 22 x 0 x2 4 / 5 1 25 x 2 16 x 2 or 5 2 x 3 x 3 3 x 5 x ~ c31 x 0 x2 1 36 9 7 x 2 20 x 3 or 6 3 x 4 x 4 4 x 6 x ~ a32 x 0 x2 4 / 5 1 25 x 2 16 x 2 or 5 2 x 3 x 3 3 x 5 x ~ a 41 x 0 x2 4 / 5 1 25 x 2 16 x 2 or 5 2 x 3 x 3 3 x 5 x ~ a42 x 0 x 2 0.25 / 0.75 1 4 x2 3 x 0.5 or 2 0. 5 x 1 x 1 1 x 2 x ~ b1 x

0 x 2 400 41 1 529 x 3 88 x 20 or 23 20 x 21 x 21 21 x 23 x ~ b2 x 0 x 2 676 53 1 841 x 2 112 x 26 or 29 26 x 27 x 27 27 x 29 x ~ b3 x 0 x 2 1936 / 89 1 2209 x 2 184 x 44 or 47 44 x 45 x .45 45 x 47 x ~ b4 x 0 x 2 841 / 59 1 1024 x 2 124 x 29 or 32 29 x 30 x .30 30 x 32 x

Fuzzy MODM Models and Approaches 515 Stage 1: Initialization ~ ~ ~ Step 1. Input membership functions of c for object ive functions f x c x , ~ ~ ~ ~ a and b for constraints a x b . We set an initia l satisfactory degree as 0.2. We use default values for the weights of objective functions. = 0.2, we calculate the Pareto optimal Steps 2–4. Under the degree sol ution. Associated with the FMOLP problem in this example, a corresponding MOLP p roblem is listed: 9 max 9 3 1 x1 x2 36 20 6.25 2.25 3 1 16 12 9 9 36 20 subject to 4 3 0.75 0.75 4 3 9 9 0.25 0.25 5 4 41 400 25 16 5 4 x1 x1 529 88 53 676 25 16 5 4 841 112 89 1936 36 20 5 4 25 16 0.75 4 3 0.25 2209 184 59 841 25 16 1024 124 where ,1 . Refer to the MOLP problem, initially 0 0.2 and 1 1 ; then 8 non fuzzy objective functions and 16 non fuzzy constraints are generated. The result is listed as fo llows: 10.8 18 32 max 4 5.8 2 1.6 2 4 2 10.8 18 32 2 13.6 x1 x2 1.6

516 J. Lu et al. subject to 3.4 1 0.4 1 0.4 1 3.4 1 10.8 18 32 4 5 3 21.8 3 1 1 3.4 3 0.4 3 21.8 3 5 3 21.8 x1 x1 3 5 3 21.8 5 408.2 21 501.6 21 686.6 27 818.6 27 1953.8 45 2245.8 45 852.8 30 999.2 30 The interval [ , 1] is further split. We then have * x1 1 .9115 5.1023 x* 2 and two optimal objective values ~* * * f 1 x1 , x 2 ~* f 1 1.9115 ,5.1023 ~ 1.9115 c 11 ~ 5.1023 c 12 ~* * * f 2 x1 , x 2 ~* f 2 1.9115 ,5.1023 ~ 1.9115 c 21 5.1023 ~22 . c Steps 5 and 6. Suppose the decision maker is not satisfied with the initial Pare to optimal solution; the interactive process will start.

Fuzzy MODM Models and Approaches 517 Stage 2: Iterations Iteration No. 1: Step 7. Based on the Pareto optimal solutio n obtained in Stage 1, the decision ~ ~ maker specifies new fuzzy goals g 1 , g 2 by increasing 30% of the value of the ~* * * first objective function f 1 x 1 , x 2 ~1* 1.9115 ,5.1023 1.9115 ~11 5.1023 ~12 and f c c ~ * ~ decreasing 25% of the value of the second one f 2* x 1 , x * f 2* 1.9115 ,5.1023 2 ~ c 1.9115 c 2 1 5.1023 ~22 . That is, ~ ~ g1 , g2 ~ * ~ * 1.3 * f 1* x 1 , x * , 0.75 * f 2* x 1 , x * 2 2 Steps 8–10. Calculate the fuzzy Pareto optimal solution based on the new ~ ~ fuzzy goals g 1 , g 2 and the satisfactory degree = 0.2. Under the new fuzzy goals, t he FMOLP problem is converted into a nonfuzzy MOLP m problem as follows: min max 9 9 36 20 6.25 2.25 3 1 3 1 16 12 9 9 36 20 2.4849 2.4849 1.5292 1.5292 9 9 6.6329 3 1 36 20 6.6329 16 12 6.25 2.25 4.0818 9 9 3 1 4.0818 36 20 x1 x2 subject to 4 3 0.75 0.25 0.75 0.25 4 3 9 9 36 20 5 4 25 16 5 4 25 16 5 4 25 16 5 4 25 16 0. 75 0.25 4 3 41 529 53 841 89 2209 59 1024 400 88 676 112 1936 184 841 124 x1 x1 where We obtain ,1 ; * x1 3 .0486 x* 2 4 .9239 ,

518 J. Lu et al. and two optimal fuzzy objective values are ~* * * f 1 x1 , x2 ~* * * f 2 x1 , x2 ~* f 1 3.0486 ,4.9239 ~* f 2 3.0486 ,4.9239 ~ 3.0486 c11 ~ 3.0486 c21 c 4.9239 ~12 ~ 4.9239 c22 . Comparing the two groups of objective values, we can find that the first fuzzy o bjective function has some improvement, and the second one has some decrement. S tep 11. Suppose the decision maker does not satisfy the fuzzy Pareto optimal sol ution, the interactive process will proceed; that is, start the second iteration . Iteration No. 2: Step 7. At this iteration, suppose the decision maker specifi es new fuzzy ~ ~ goals g 1 , g 2 by the corresponding membership functions as fo llows: 0 x ~ g1 2 x 196 245 14 or 37 x 21 x 37 21 x 14 x x 1 1369 x 2 928 21 0 x ~ g2 2 x 42.25 114 6.5 or 25 x 12.5 x 25 12.5 x 6.5 x x 1 625 x 2 468.75

12.5 Steps 8–10. Calculate the fuzzy Pareto optimal solution based on the new ~ ~ fuzzy goals g 1 , g 2 , and keep the degree = 0.2. Under the fuzzy goals, the FMOLP p roblem is converted into the non fuzzy MOLP m problem as follows:

Fuzzy MODM Models and Approaches min max 9 9 36 20 6.25 2.25 3 1 3 1 16 12 9 9 36 20 245 196 1369 928 114 42.25 6 25 468.75 519 x1 x2 subject to 4 3 0.75 0.25 0.75 0.25 4 3 9 9 36 20 5 4 25 16 5 4 25 16 5 4 25 16 5 4 25 16 0. 75 0.25 4 3 41 529 53 841 89 2209 59 1024 400 88 676 112 1936 184 841 124 x1 x1 where We have ,1 . * x1 2.8992 x* 2 4.9829 and two optimal objective values are ~* * * f 1 x1 , x 2 ~* * * f 2 x1 , x 2 ~* f 1 2.8992,4.9829 ~* f 2 2.8992 ,4.9829 2.8992~11 c 2.8992 ~21 c 4.9829~12 c 4.9829 ~22 . c Step 11. Now the decision maker is satisfied with the fuzzy Pareto optimal solut ion obtained in Step 10; the interactive process thus terminates. The current fu zzy Pareto optimal solution is the final satisfactory solution of the FMOLP prob lem to the decision maker as follows: * x1 * 2.8992 x 2 4.9829 ~* * * ~* f 1 x1 , x 2 f 1 2.8992 ,4.9829 ~ 2.8992 c 11 ~ 2.8992 c 21 ~ 4.9829 c 12 ~ 4.9829 c 22 ~* * * f 2 x1 , x 2 ~* f 2 2.8992 ,4.9829 .

520 J. Lu et al. This example illustrates the proposed fuzzy-goal fuzzy multi-objective decisionmaking method. 7. CONCLUSION This chapter presented a set of models and an interactive method to describe and solve the FMOLP problems. In the proposed FMOLP models, fuzzy parameters can ap pear in both objective functions and constraints and can be described by any for m of membership function. When only objective functions or only constraints incl ude fuzzy parameters, the model is still as an FMOLP problem since a real number is a special case of a fuzzy number. Similarly, a goal of a decision maker with a real number is also a special case of a fuzzy goal in the models. The propose d FMODM method extends MODM decision analysis functions from a crisp to an impre cise scope and improved existing FMODM methods. It allows decision makers to exp ress their goals by any form of membership function. When decision makers do not have a clear idea to how of choose a suitable form of membership function, they can try different forms. This feature offers decision makers a much higher conf idence in using the method to solve their practical problems. A decision support system has been developed to apply the method to assist decision makers to solv e realistic FMOLP problems. This system has been initially tested by a number of examples, and results are very positive for our research project supported by t he Australian Research Council (ARC). ACKNOWLEDGMENTS This research is partially supported by the Australian Research Council (ARC) un der Discovery Grant DP0211701. REFERENCES Bellman, R.E., and Zadeh, L.A., 1970, Decision-making in a fuzzy environment, Ma nagement Science, 17: 141 164. Carlsson, C., and Fuller, R., 1996, Fuzzy multipl e criteria decision making: recent developments, Fuzzy Sets and Systems, 78: 139 152.

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FUZZY OPTIMIZATION VIA MULTIOBJECTIVE EVOLUTIONARY COMPUTATION FOR CHOCOLATE MAN UFACTURING Fernando Jiménez1, Gracia Sánchez1, Pandian Vasant2, and José Luis Verdegay3 1 Department of Ingeniería de la Información y las Comunicaciones, University of Murci a, Spain 2Universiti Teknologi Petronas, Malaysia 3Department of Ciencias de la Computaci´on e Inteligencia Artificial, University of Granada, Spain Abstract: This chapter outlines, first, a real-world industrial problem for product mix se lection involving 8 variables and 21 constraints with fuzzy coefficients and, se cond, a multi-objective optimization approach to solve the problem. This problem occurs in production planning in which a decision maker plays a pivotal role in making decisions under a fuzzy environment. Decision maker should be aware of h is/her level-of-satisfaction as well as degree of fuzziness while making the pro duct mix decision. Thus, the authors have analyzed using a modified S-curve memb ership function for the fuzziness patterns and fuzzy sensitivity of the solution found from the multi-objective optimization methodology. An ad hoc Pareto-based multi-objective evolutionary algorithm is proposed to capture multiple nondomin ated solutions in a single run of the algorithm. Results obtained have been comp ared with the well-known multi-objective evolutionary algorithm NSGA-II. Multi-o bjective optimization, evolutionary algorithm, NSGA-II Key words: 1. INTRODUCTION It is well known that optimization problems originate in a variety of situations . Particularly interesting are those concerning management problems as decision makers usually state their data in a vague way: “high benefits,” “as low as possible,” “im portant savings,” etc. Because of this C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 523

524 F. Jiménez et al. vagueness, managers prefer to have not just one solution but a set of them, so t hat the most suitable solution can be applied according to the state of existing decision of the production process at a given time and without increasing delay . In these situations, fuzzy optimization is an ideal methodology, since it allo ws us to represent the underlying uncertainty of the optimization problem, while finding optimal solutions that reflect such uncertainty and then applying them to possible instances, once the uncertainty has been solved. This allows us to o btain a model of the behavior of the solutions based on the uncertainty of the o ptimization problem. Fuzzy constrained optimization problems have been extensive ly studied since the 1970s. In the linear case, the first approaches to solve th e so called fuzzy linear programming problem appeared in Bellmann and Zadeh (197 0), Tanaka et al. (1974), and in Zimmerman (1976). Since then, important contrib utions solving different linear models have been made and these models have been the subject of a substantial amount of work. In the nonlinear case (Ali, 1998; Ekel et al., 1998; Ramik and Vlach, 2002) the situation is quite different, as t here is a wide variety of specific and both practically and theoretically releva nt nonlinear problems, with each having a different solution method. In this cha pter a real-life industrial problem for product mix selection involving 21 const raints and 8 variables has been considered. This problem occurs in production pl anning in which a decision maker plays a pivotal role in making decisions under a highly fuzzy environment. Decision maker should be aware of his/her level-of-s atisfaction as well as degree of fuzziness while making the product mix decision . Thus, the authors have analyzed using the sigmoidal membership function, the f uzziness patterns and fuzzy sensitivity of the solution. In Vasant (2003, 2004, 2006) a linear case of the problem is solved by using a linear programming itera tive method that is repeatedly applied for different degrees of satisfaction val ues. In this chapter, a nonlinear case of the problem is considered and we propo se a multi-objective optimization approach in order to capture solutions for dif ferent degrees of satisfaction with a simple run of the algorithm. This multi-ob jective optimization approach has been proposed by Jiménez et al. (2004a, 2004b, 2 006) within a fuzzy optimization general context. Given this background, this ch apter is organized as follows: In section 2 a nonlinear case study in a chocolat e manufacturing firm is described, and its mathematical formulation is stated. S ection 3 we propose a multiobjective optimization approach for this problem and an ad hoc multiobjective evolutionary algorithm. Section 4 shows results obtaine d with the proposed multi-objective evolutionary algorithms and the well-known N SGA-II algorithm. Finally, Section 5 offers the main conclusions and future rese arch.

Multi-objective Evolutionary Computation 525 2. NONLINEAR CASE STUDY IN A CHOCOLATE MANUFACTURING FIRM Due to limitations in resources for manufacturing a product and the need to sati sfy certain conditions in manufacturing and demand, a problem of fuzziness occur s in industrial systems. This problem occurs also in chocolate manufacturing whe n deciding a mixed selection of raw materials to produce varieties of products. This is referred here to as the product mix selection problem (Tabucanon, 1996). There are a number of products to be manufactured by mixing different raw mater ials and using several varieties of processing. There are limitations in resourc es of raw materials and facility usage for the varieties of processing. The raw materials and facilities usage required for manufacturing each product are expre ssed by means of fuzzy coefficients. There are also some constraints imposed by the marketing department such as product mix requirement, main product line requ irement, and lower and upper limit of demand for each product. It is necessary t o obtain maximum profit with a certain degree of satisfaction of the decision ma ker. 2.1 Fuzzy Constrained Optimization Problem The firm Chocoman Inc. manufactures eight different kinds of chocolate products. Input variables xi represent the amount of manufacturated product in 103 units. The function to maximize is the total profit obtained calculated as the summati on of profit obtained with each product and taken into account the applied disco unt. Table 1 shows the profit (ci) and discount (di) for each product i. Table 1. Profit (ci) and Discount (di) in $ per 103 units Product (xi) x1 = Milk chocolate, 250 g x2 = Milk chocolate, 100 g x3 = Crunchy chocolate, 250 g x4 = Crunchy chocolate, 100 g x5 = Chocolate with nuts, 250 g x6 = Chocolate with nut s, 100 g x7 = Chocolate candy x8 = Chocolate wafer Synonym MC 250 MC 100 CC 250 CC 100 CN 250 CN 100 CANDY WAFER Profit (ci) c1 = 180 c2 = 83 c3 = 153 c4 = 72 c 5 = 130 c6 = 70 c7 = 208 c8 = 83 Discount (di) d1 = 0.18 d2 = 0.05 d3 = 0.15 d4 = 0.06 d5 = 0.13 d6 = 0.14 d7 = 0.21 d8 = 0.1 The lower limit of demand for each product i is 0 in all cases, whereas the uppe r limit (ui) is shown in Table 2.

526 Table 2. Demand (ui) in $ per 103 Units Product MC 250 MC 100 CC 250 CC 100 CN 2 50 CN 100 CANDY WAFER Demand (ui) u1 = 500 u2 = 800 u3 = 400 u4 = 600 u5 = 300 u 6 = 500 u7 = 200 u8 = 400 F. Jiménez et al. There are eight raw materials to be mixed in different proportions and nine proc esses (facilities) to be utilized. Therefore, there are 17 constraints with fuzz y coefficients separated in two sets such as raw material availability and facil ity capacity. These constraints are inevitable for each material and facility th at is based on the material consumption, facility ~ usage, and the resource avai lability. Table 3 shows fuzzy coefficients a ij l h represented by a ij ,a ij fo r required materials and facility usage j for manufacturing each product i and n onfuzzy coefficients bj for availability of material or facility j. Table 3. Raw Material and Facility Usage Required (per 103 units) a ij ,a ij and Availability (bj ) MC 250 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 66, 109 47, 78 0, 0 75, 125 0, 0 375, 625 337, 562 45, 75 0.4, 0.6 0, 0 0.6, 0.9 0, 0 0, 0 0.07, 0.12 0.2, 0.3 0.04, 0.06 0.2, 0.4 MC 100 26, 44 19, 31 0, 0 30, 50 0, 0 0, 0 0, 0 95, 150 0.1, 0.2 0, 0 0.2, 0.4 0, 0 0, 0 0.07, 0.12 0, 0 0.2, 0.4 0.2, 0.4 CC 250 56, 94 37, 62 28, 47 66, 109 0, 0 375, 625 337, 563 45, 75 0.3, 0.5 0.1, 0.2 0.6, 0.9 0.2, 0.3 0, 0 0.07, 0.12 0.2, 0.3 0.04, 0.06 0.2, 0.4 CC 100 22, 37 15, 25 11, 19 26 , 44 0, 0 0, 0 0, 0 90, 150 0.1, 0.2 0.04, 0.07 0.2, 0.4 0.07, 0.12 0, 0 0.07, 0 .12 0, 0 0.2, 0.4 0.2, 0.4 CN 250 37, 62 37, 62 56, 94 56, 94 0, 0 0, 0 337, 562 45, 75 0.3, 0.4 0.2, 0.3 0.6, 0.9 0, 0 0, 0 0.07, 0.12 0.2, 0.3 0.04, 0.06 0.2, 0.4 CN 100 15, 25 15, 25 22, 37 22, 37 0, 0 0, 0 0, 0 90, 150 0.1, 0.2 0.07, 0. 12 0.2, 0.4 0, 0 0, 0 0.07, 0.12 0, 0 0.2, 0.4 0.2, 0.4 CANDY 45, 75 22, 37 0, 0 157, 262 0, 0 0, 0 0, 0 1200, 2000 0.4, 0.7 0, 0 0.7, 38718 0, 0 0, 0 0.15, 0.2 5 0, 0 1.9, 3.1 1.9, 3.1 WAFER 9, 21 9, 21 0, 0 18, 30 54, 90 187, 312 0, 0 187, 312 0.1, 0.12 0, 0 0.3, 0.4 0, 0 0.2, 0.4 0, 0 0, 0 0.1, 0.2 1.9, 3.1 Availabil ity 100,000 120,000 60,000 200,000 20,000 500,000 500,000 500,000 1000 200 1500 200 100 400 400 1200 1000 l h

Multi-objective Evolutionary Computation 527 Material or Facility Cocoa (kg), Milk (kg), Nuts (kg), Cons.sugar (kg), Flour (k g), Alum.foil (ft2), Paper(ft2), Plastic (ft2), Cooking(ton-hours), Mixing(ton-h ours), Forming(ton-hours), Grinding(ton-hours), Wafer making(ton-hours), Cutting (hours), Packaging 1(hours), Packaging 2(hours), Labor(hours) Additionally, the following constraints were established by the sales department of Chocoman Inc.: 1. Main product line requirement. The total sales from candy and wafer products should not exceed 15% of the total revenues from the chocolate bar products. Ta ble 4 shows the values of sales/revenues (ri) for each product i. 2. Product mix requirements. Large-sized products (250 g) of each type should not exceed 60% o f the small-sized product (100 g). Table 4. Revenues/Sales (ri) in $ per 103 Units Product MC 250 MC 100 CC 250 CC 100 CN 250 CN 100 CANDY WAFER Revenues/Sales (ri) r1 = 375 r2 = 150 r3 = 400 r4 = 160 r5 = 420 r6 = 175 r7 = 400 r8 = 150 2.2 Membership Function for Coefficients We consider the modified S-curve membership function proposed by ~ Vasant (2003) . For a value x, the degree of satisfaction, aij ( x ) for fuzzy ~ij is given by the membership function given in (1). coefficient a

528 F. Jiménez et al. 1.00 0.999 aij x x B l x aij l aij l aij ( x) l aij x h aij h aij h aij (1) 1 Ce 0.001 0.000 h l aij aij x x Given a degree of satisfaction value µ, the crisp value aij µ for fuzzy ~ coefficien t aij can be calculated using Eq. (2). l aij h aij l aij aij ln 1 B 1 C (2) The value determines the shape of the membership function, whereas B and C value s can be calculated from , Eqs. (3) and (4). C 0.998 0.999 0.001e B 0.999(1 C ) (3) (4) If we wish that for a degree-of-satisfaction value µ = 0.5, the crisp value aij 0 .5 is in the middle of the interval aij 0.5 then, = 13.81350956. l aij

h aij 2

Multi-objective Evolutionary Computation 529 2.3 Problem Formulation Given a degree of satisfaction value µ, the fuzzy constrained optimization problem can be formulated (Jiménez et al., 2006; Vasant, 2004) as the non linear constrai ned optimization problem shown in the following: 8 Maximize i 1 ci xi d i xi 2 subject to 8 l aij i 1 8 6 h aij l aij ln 1 B 1 xi C 0 bj 0, j 1,..., 17 ri xi i 7 0.15 i 1 ri xi x1 x3 x5 0 0.6 x2 0.6 x4 0.6 x6 xi ui , 0 0 0 i 1,..., 8 3. A MULTI-OBJECTIVE EVOLUTIONARY APPROACH In this section, we propose a multi-objective optimization approach to solve the problem shown above for all satisfaction degree values, which composes the fuzz y solution of the former fuzzy optimization problem. In the multi-objective opti mization problem, a new input variable is considered in order to find the optima l solution for each degree-ofsatisfaction value (Jimenez et al., 2004a, 2004b, 2

006). The following formulation shows the multi-objective constrained optimizati on problem for Chocoman Inc. In this problem, x9 represents the degree-of-satisf action value, which must be minimized to generate the desired Pareto front.

530 8 F. Jiménez et al. M aximize i 1 ci xi x9 h a ij l a ij d i xi 2 M inimize subject to 8 l a ij i 1 8 6 ln 1 C 0 B 1 xi bj 0, j 1, ..., 17 ri xi i 7 0.15 i 1 ri xi x1 x3 x5 0 0.6 x 2 0.6 x 4 0.6 x 6 xi ui , x9 0 0 0 i 1, ..., 8 0.999, 0.001 Multi-objective Pareto-based evolutionary algorithms (Coello et al., 2002; Deb, 2001; Jiménez et al., 2002) are especially appropriate to solve multi-objective no nlinear optimization problems because they can capture a set of Pareto solutions in a single run of the algorithm. We propose an ad hoc multi-objective Pareto-b ased evolutionary algorithm to solve the Chocoman Inc. problem. The algorithm us es a realcoded representation, uniform and arithmetical cross, and uniform, nonu niform and minimal mutation (Jiménez et al., 2002). Diversity among individuals is maintained by using an ad hoc elitist generational replacement technique. The a lgorithm has a population P of N solutions. For each solution i, f ji is the val ue for the j-th objective (j = 1, . . . , n) and g ij is the value for the j-th constraint (j = 1, . . . , m). For the Chocoman Inc. problem, n = 2 and m = 21.

Given a population P of N individuals, N children are generated by random select ion, crossing, and mutation. Parents and children are ordered in N slots in the following way. A solution i belongs to slot si such that si f i2 N The order inside slots is established with the following criteria. Position pi o f solution i is lower than position pj of solution j in the slot if: i is feasib le and j is unfeasible, i and j are unfeasible and gimax gjmax, i and j are feas ible and i dominates j

Multi-objective Evolutionary Computation 531 i and j are feasible and nondominated and cdi > cdj where gimax = maxj = 1,...,m {gij} and cdi is a metric for the crowding distance of solution i: , cdi n if f f supi j max j f ji fi max or f ji fi min for any i f f j 1 in i j min j , in anothercase where f max j max i j f i j f min j min f i j i 1 ,..., N i 1 ,..., N is the value of the jth objective for the higher solution adjacent in the jth ob

jective to i, f sup j f is the value of the jth objective for the solution lower adjacent in the jth o bjective to i. f i inf j j The new population is obtained by selecting the N best individual from the paren t and children. The following heuristic rule is considered to establish an order . Solution i is better than solution j if: pi < pj pi = pj and cdi > cdj where p i is the position of solution i in its slot. 4. EXPERIMENTS AND RESULTS To compare performance of the algorithms in multi-objective optimization, we hav e followed an empirical methodology similar to that proposed in Laumanns et al. (2001) and Purshouse and Fleming (2002). It has been used as a measure that calc ulates the fraction of the space that is not dominated by any of the solutions o btained by the algorithm (Laumanns et al., 2001; Zitler et al., 2003). The aim i s to minimize the value of . This measure estimates both the distance of solutio ns to the real Pareto front and the spread. Value can be calculated as shown in Eq. (5) where Pı is composed

532 F. Jiménez et al. by the Nı non dominated solutions of P and f ju max and f ju min are the utopia ma ximum and minimum value for the j-th objective. For the Chocoman Inc. problem, u topia minimum and maximum values are shown in Table 5. Nı n 1 f 1 i 1 u max n n f f i n j 1 fj f supij f ji (5) u max j u min j j 1 Table 5. Utopia Minimum and Maximum Values for the Chocoman Inc. Problem. Utopia Min. 140,000 0.001 Utopia Max. 200,200 0.999 Objective 1 Objective 2 The parameters were set up using a previous process using a methodology similar to the one proposed in Laumanns et al. (2001). Table 6 shows the parameters obta ined. Table 6. Parameters in the Run of the Proposed Algorithm and NSGA-II for the Cho coman Inc. Problem. Number of iterations Population size Cross-probability Mutat ion probability Uniform cross-probability Uniform mutation probability Parameter c for nonuniform mutation T = 10000 N = 100 pCross = 0.8 pMutate = 0.5 pUniform Cross = 0.7 pUniformMutate = 0.7 c = 2.0 Various metrics for both convergence and diversity of the populations obtained h ave been proposed for a more exact evaluation of the effectiveness of the evolut ionary algorithms. In his book, Deb (2001) assembles a wide range of the metrics that figure in the literature. For this chapter we propose the use of two metri cs to evaluate the goodness of the algorithm. The first metric we use is the gen erational distance ( ) proposed by Veldhuizen and Lamont (1999), evaluates the p roximity of the population to the Pareto optimal front by calculating the averag e distance of the population from an ideal population P* made up of N* solutions distributed uniformly along the Pareto front. This metric is shown in Eq. (6).

Multi-objective Evolutionary Computation N 533 r i 1 di v N (6) We use v = 1, and parameter dmini is the Euclidean distance (in the objective sp ace) between the solution i and the nearest solution in P*: di min k 1 N * n j 1 f ji f *k j 2 where f j* k k is the value of the j-th objective function for the k-th solution in P*. For our problem, we use the points in Tables 7 and 8 as the ideal popula tion P*. Table 7. Optimal Points for Uniformly Distributed Values In µ Obtained with Gradie nt for the Chocoman Inc. Problem µ 0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.999 x1 2.397.161 2.794.339 2.875.612 2.932.170 2.980.141 3.025.503 3.072.207 3.124. 697 3.191.108 3.296.295 4.143.502 x2 3.995.268 4.657.232 4.792.687 4.886.950 4.9 66.902 5.042.506 5.120.344 5.207.829 5.318.513 5.493.824 6.905.837 x3 1.989.859 2.347.411 2.420.906 2.472.111 2.515.579 2.556.712 2.599.091 2.646.756 2.707.111 2.802.814 3.540.144 x4 3.316.432 3.912.352 4.034.844 2.472.111 4.192.632 4.261.1 87 4.331.819 4.411.260 4.511.852 4.671.356 5.900.239 x5 1.411.270 1.553.713 1.58 3.183 1.603.752 1.621.237 1.637.803 1.654.891 1.674.136 1.698.541 1.737.323 2.00 0.325 Table 8. Optimal Points for Uniformly Distributed Values in µ Obtained with Gradie nt for the Chocoman Inc. Problem-(Continued) µ 0.001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 .8 0.9 0.999 x6 2.352.116 2.589.522 2.638.638 2.672.920 2.702.061 2.729.672 2.75 8.152 2.790.226 2.830.901 2.895.538 3.333.874 x7 1.392.046 1.593.681 1.635.240 1 .664.218 1.688.830 1.712.133 1.736.153 1.763.184 1.797.432 1.851.788 2.000.000 x 8 1.170.292 1.673.137 1.772.523 1.841.035 1.898.738 1.952.965 2.008.459 2.070.43 4 2.148.254 2.270.207 5.448.504 Profit 150089.2 165662.6 168585.9 170566.9 17221 2.8 173740.0 175282.7 176980.6 179074.1 182264.4 200116.4

534 F. Jiménez et al. The second metric we use is the spread ( ) put forward by Deb (2001) to evaluate the diversity of the population. Equation (7) shows this measure. n j 1 dj n e N i 1 e dj di Nd d (7) j 1 where di may be any metric of the distance between adjacent solutions, and d is the mean value of such measurements. In our case, di has been calculated using t he Euclidean distance. Parameter d e is the distance j between the extreme solut ions in P* and P corresponding to the j-th objective function. Table 9 shows the best, worst, medium, and variance values for the , , and measures obtained in 1 0 executions of both algorithms. Table 9. Results of 10 Runs of the Proposed Algorithm and NSGA-II for the Chocom an Algorithm Proposed algorithm NSGA-II Proposed algorithm NSGA-II Proposed algo rithm NSGA-II best 0.5366 0.5519 Ybest 227781.6632 2.27914.8763 best 0.9737 0.97 35 worst 0.583 0.5928 Yworst 228187.1852 2.28427.1933 worst 0.9898 0.9809 mean 0 .5589 0.5715 Ymean 2.28031.9.239 228228.772 mean 0.9837 0.9784 variance 2.1568 × 1 0-5 1.143 × 10-5 Yvariance 1479.6619 2724.6756 variance 2.8096 × 10-6 6.0036 × 10-7 Figure 1 shows the non dominated solutions obtained in the best of 10 executions of the proposed algorithm and NSGA-II for the Chocoman Inc. problem. 5. 5.1 CONCLUSIONS AND FUTURE WORKS Conclusions Fuzzy nonlinear optimization problems are, in general, difficult to solve. In th is chapter we describe a multi-objective approach to solving a fuzzy

Multi-objective Evolutionary Computation 535 nonlinear constrained optimization problem that appears in production planning f or chocolate manufacturing. A Pareto-based evolutionary algorithm is proposed to capture the solution in a single run of the algorithm. Optimality and diversity metrics have been used for the evaluation of the effectiveness of the proposed multi-objective evolutionary algorithm compared with the well-known algorithm NS GA-II. We show the values obtained using these metrics for the solutions generat ed by both algorithms. The results show a real ability of the proposed approach to solve problems in production planning for chocolate manufacturing. Chocoman Inc. problem 200000 Proposed Algorithm 190000 NSGA-II 180000 Profit 170000 160000 150000 140000 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Degree of Satisfaction Figure 1. Nondominated solutions obtained with the proposed algorithm and NSGAII for the Chocoman Inc. problem. 5.2 Future Works Multi-objectives with several other objective functions can be considered for fu ture research work as well as fuzzy costs and fuzzy right-side coefficients in c onstraints. There is a possibility of designing a productive, computational inte lligence, self-organized evolutionary fuzzy system.

536 F. Jiménez et al. ACKNOWLEDGMENTS Research supported in part by FEDER funds under grants MINAS (TIC00129-JA) and H euriFuzzy (TIN2005-08404-C04-01) REFERENCES Ali, F.M., 1998, A differencial equarion approach to fuzzy non-linear programmin g problems, Fuzzy Sets and Systems, 93(1): 57 61. Bellman, R.E., Zadeh, L.A., 19 70, Decision Making in a fuzzy environment, Management Science, 17: 141 164. Coe llo, C.A., Veldhuizen, D.V., Lamont, 2002, G.V., Evolutionary Algorithms for Sol ving Multi-Objective Problems, Kluwer Academic/Plenum publishers, New York. Deb, K., Agrawal, S., Pratap, A., Meyarivan, T., 2000, A fast elitist nondominated s orting genetic algorithm for multi-objective optimization: NSGAII, In: Proceedin gs of the Parallel Problem Solving from Nature VI (PPSN-VI), pp. 849 858. Deb, K ., 2001, Multi-Objective Optimization using Evolutionary Algorithms, John Wiley and Sons, New York. Ekel, P., Pedrycz, W., Schinzinger, R., 1998, A general appr oach to solving a wide class of fuzzy optimization problems, Fuzzy Sets and Syst ems, 97(1): 49 66. Jiménez, F., Gómez-Skarmeta, A.F., Sánchez, G., Deb, 2002, K., An e volutionary algorithm for constrained multi-objective optimization, Proceedings IEEE World Congress on Evolutionary Computation. Jiménez, F., Gómez-Skarmeta, A.F, Sán chez, G., 2004, A multi-objective evolutionary approach for nonlinear constraine d optimization with fuzzy costs, IEEE International Conference on Systems, Man & Cybernetics (SMC’04) The Hague, Netherlands. Jiménez, F., Gómez-Skarmeta, A.F, Sánchez, G., 2004, Nonlinear optimization with fuzzy constraints by multi-objective evol utionary algorithms, Advances in Soft Computing. Computational Intelligence, The ory and Applications, pp. 713 722 Jiménez, F., Cadenas, J.M., Sánchez, G., Gómez-Skarm eta, A.F., Verdegay, J.L., 2006, Multi-objective evolutionary computation and fu zzy optimization, International Journal of Approximate Reasoning, 43: 59-75. Lau manns, M., Zitzler, E., and Thiele, L., 2001, On the effects of archiving, eliti sm, and density based selection in evolutionary multi-objective optimization, Pr oceedings of the First International Conference on Evolutionary Multi-Criterion Optimization (EMO 2001), Zitzler, E., et al. (eds.), pp. 181 196. Purshouse, R.C ., Fleming, P.J., 2002, Why use elitism and sharing in a multiobjective genetic algorithm, Proceedings of the Genetic and Evolutionary Computation Conference, p p. 520 527. Ramik, J., Vlach, M., 2002, Fuzzy mathematical programming: a unifie d approach based on fuzzy relations, Fuzzy Optimization and Decision Making, 1: 335–346. Tabucanon, T.T., 1996, Multi objective programming for industrial enginee rs, Mathematical Programming For Industrial Engineers, pp. 487–542, Marcel Dekker, Inc., New York.

Multi-objective Evolutionary Computation 537 Tanaka, H., Okuda, T., Asai, K., 1974, On fuzzy mathematical programming, Journa l of Cybernetics, 3: 37 46. Vasant, P., 2003, Application of fuzzy linear progra mming in production planning, Fuzzy Optimization and Decision Making, 2(3): 229 241. Vasant, P., 2004, Industrial production planning using interactive fuzzy li near programming, International Journal of Computational Intelligence and Applic ations, 4(1): 13 26. Vasant, P., 2006, Fuzzy production planning and its applica tion to decision making, Journal of Intelligent Manufacturing, 17(1): 5 12. Veld huizen, D.V., Lamont, G.B., 1999, Multiobjective evolutionary algorithms: classi fications, analyses, and new innovations, Ph.D. thesis, Air Force Institute of T echnology. Technical Report No. AFIT/DS/ENG/99 01, Dayton, Ohio: Zimmermann, H.J ., 1976, Description and optimization of fuzzy systems, International Journal of General Systems, 2: 209–215. Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M. , Grunert da Fonseca, V., 2003, Performance assessment of multiobjective optimiz ers: an analysis and review, IEEE Transactions on Evolutionary Computation, 7(2) : 117 132.

MULTI-OBJECTIVE GEOMETRIC PROGRAMMING AND ITS APPLICATION IN AN INVENTORY MODEL Tapan Kumar Roy Department of Mathematics, Bengal Engineering and Science University, Shibpur Ho wrah, West Bengal, India Abstract: In this chapter, first the general multi-objective geometric programming problem is defined, then Pareto optimality, the fuzzy geometric programming technique t o solve a multi-objective geometric programming problem is discussed, and finall y a multi-objective marketing planning inventory problem is explained and formul ated. Numerical examples are given for the inventory problem in a multinational soft drink manufacturing company. Multi-objective, geometric programming, fuzzy sets, inventory, Pareto optimality, posynomial function Key words: 1. INTRODUCTION As society becomes more complex and as the competitive environment develops, bus iness persons are finding that they require multiple objectives. Almost every im perative real-world problem involves more than one objective. In such cases, dec ision makers evaluate the best possible approximate solution alternatives accord ing to multiple criteria. A general multiple objective (or multiple criteria) no nlinear programming (MONLP) problem is of the following form: Find x = (x1, x2,…, xn)T( * ) C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 539

540 T.K. Roy which minimizes F(x) = (f1(x), f2(x) … fk(x) )T subject to g j ( x ) and bj , ( j 1, 2,..., m ) x( ( x1 , x2 ,...xk )T ) 0 where f1 ( x), f2 ( x),.... f k ( x) are ( 2) and g j (x) ( j 1, 2,...., m) are functi ons. Here fi : Rn R for i 1,2,...., k and g j : Rn R for j 1,2,..., m REMARK 1. When k = 1, problem (*) reduces to a single objective NLP problem. It is noted that if the objectives of the original problem are to minimize f r ( x) for r k0 1, k0 2,....k then the objective in the mathematical formulation will be Minimize fk 0 2 F ( x) , f k ( x)) ( f1 ( x), f 2 ( x) T , f k0 ( x), f k0 1 ( x), ( x), subject to the same constraints as in (*) If f r ( x), (r 1, 2, , k ), g j ( x), ( j 1, 2, , m) are linear, the corresponding problem (*) is called multiple obj ective linear programming (MOLP) problem. When all or any one of the above funct ions are nonlinear, it is referred as a MONLP problem. When all of the above fun ctions are posynomial or signomial, (*) is referred as a multi-objective geometr ic programming problem (MOGPP).

Geometric Programming: An Inventory Model 541 A MOGPP can be stated as Find xt Minimize Maximize x1 , x2 , , xn f1 x T so as to T10 n (1) x ra1 ir r 1 n 0 c 10i i 1 T 20 0 c2i i 1 r 1 f2 x T k0 x ra 2 ir 0 x>0 ................................................... n Minimize subject to Tp fk x i 1 c k0i r 1 x ra k ir 0 n gp x s 1 c ps r 1

x r p sr a 1 p 1, 2 ... ...., m where c 0 ji 0 , cks 0 , a 0 , a jir are all real numbers for j = 1, jir 2,.., k; i = 1, 2, .., T j0 ; k = 1, 2, …, m; s = 1, 2, …, Tk Let X be a set of cons traints of (*) such that X xn ) T {x with n gi (x) 0 for bj , j i 1, 2, 1, 2, , m, and x , n} (x1 , x2 , , xi REMARK 2. The multi–objective optimization problem is convex if all the objective functions and the feasible region are convex. 2. PARETO OPTIMALITY In single objective optimization problems, the main focus is on the decision var iable space, whereas in the multi-objective framework, we are often more interes ted in the objective space (see Ehrogott, 2005). In multiobjective programming p roblems, multiple objectives are usually noncommensurable and cannot be combined into a single objective. In the MONLP problem, the objectives are simultaneousl y optimized. But due to an intrinsic conflicting nature among the objectives, it is not possible to

542 T.K. Roy find a single solution that would be optimal for all the objectives simultaneous ly. Consequently, the aim in solving MONLP is to find a compromise or satisfying solution of the decision maker. There is no natural ordering in the objective s pace because it is only partially ordered. For example, ( 3 ,3 )T can be said to be less than ( 7 ,7 )T , but we cannot say any such order between ( 6 ,2 )T and ( 5 ,8 )T . However, some of the objective vectors can be extracted for examina tion. These vectors are those where none of the components can be improved witho ut deterioration to at least one of the other components. This definition is usu ally called Pareto optimality, which is laid, by French-Italian economist and so ciologist Vilfredo Pareto (Aliprantis et al., 2001). DEFINITION 1. Let x* be the optimal solution of the following problem: Minimize subject to x X f r ( x) r 1, 2, ,k The point x* is known as ideal objective value and rth objective function value at x* i.e. f r ( x* ) is known as ideal objective value. DEFINITION 2. x* is sai d to be a Complete optimal solution to the MONLP problem (1) if there exists x* X such that f r ( x * ) f r ( x ), ( r 1, 2......, k ) for all x X. In general, the objective functions of the MONLP conflict with each other; a complete optima l solution does not always exist, and so the Pareto (or non dominated) optimalit y concept is introduced. DEFINITION 3. A decision vector x* X is a Pareto optima l solution if there does not exist another decision vector x X such that f r ( x ) f r ( x * ) for all r = 1, 2,…, k and f r 1 ( x ) f r 1 ( x * ) for at least on e r 1 1, 2, , k . * An objective vector F is Pareto optimal if there does not ex ist another * objective vector F(x) such that f r f r* for all r =1, 2,….k and f r 1 f r 1 * for at least one index r1 . Therefore, F is Pareto-optimal if the dec ision vector corresponding to it is Pareto optimal.

Geometric Programming: An Inventory Model 543 REMARK 3. In general, a Pareto optimal solution consists of an infinite number o f solutions. A Pareto optimal solution is sometimes called a noninferior solutio n since it is not inferior to other feasible solutions. DEFINITION 4. A decision vector x* X is a weakly Pareto optimal solution if there does not exist another decision vector x X such that f r ( x ) f r ( x * ) for all r = 1, 2,…, k. DEFINI TION 5. x* X is said to be a locally Pareto optimal solution to the MONLP if and only if there exists an r < 0 such that x* is Pareto optimal in X N ( x * , r ) ; i.e. there does not exist another x * X N ( x * , r ) such * that f i ( x ) f i ( x ) . Now, we introduce some non linear programming techniques, which have been used in this thesis to achieve at least local Pareto optimal solutions. 2.1 Method of Global Criterion In this method, the distance between some reference point and the feasible objec tive region is minimized. The decision maker has to select the reference point a nd the metric for measuring the distances. In this way, the multiple objective f unctions are transferred into a single objective function. We suppose that the w eighting coefficients r are real numbers such that r 0, 1, 2, , k and r k r r 1 1 The weighted Lp-problem for minimizing distances is stated as k r r 1 1 p Minimize L p ( f ( x )) fr ( x ) f r ( x* ) (2) subject to x X , for 1 p p

544 T.K. Roy 2.2 Hybrid Method Following Chankong and Haimes (1983), the hybrid problem combining Lp and the -c onstraint method is as follows: k 1/ p r r 1 * p Minimize Lp ( f ( x)) subject to f r r f r ( x) fr ( x ) (3) x where X , for 1 r p k 0, r 1, 2, , k r 1 r 1 and r ( f r ( x* )) is a real number for all r = 1, 2, …, k. k r 1 r fr ( x ) For p = 1, L1( f ( x )) f r ( x* ) (4) The objective function L1 ( f ( x)) is the sum of the weighted deviations, which

is to be minimized and is known as weighted sum method. k 1/ 2 r r 1 For p = 2, L2 ( f ( x)) f r ( x) fr ( x ) * 2 (5) When p becomes larger, the minimization of the deviation becomes more and more i mportant. , the only thing that matters is the weighted Finally, when p relative deviation of one objective function; i.e., k L ( f ( x)) r Max 1, 2, , k r r 1 f r ( x) f r ( x* ) (6) This multi-objective method is called the “min–max” method or the Tchebycheff method. Problem (6) is nondifferentiable like its unweighted

Geometric Programming: An Inventory Model 545 counterpart. Correspondingly, it can be solved in a differentiable form as long as the objective and the constraint functions are differentiable and f r ( x* ) is known globally. In this case, instead of problem (6), the problem becomes Minimize subject to x X, r fr ( x ) f r ( x* ) for all r = 1, 2, …, k (7) THEOREM 6. The solution of weighted Lp–problem (when 1 p < ) is a Pareto optimal s olution if all the weighting coefficients are positive. THEOREM 7. The solution of a weighted Tchebycheff problem (L ) is weakly Pareto optimal if all the weigh ting coefficients are positive. THEOREM 8. The weighted Tchebycheff problem has at least one Pareto optimal solution. THEOREM 9. Let a decision vector x* Minimize k r 1 X be given, Solve the problem fr ( x ) (8) subject to fr (x) fr (x* ) for all r 1, 2, , k and x 0. Let Ø(x*) be the optimal objective value. The decision vector x * is Pareto optima l if and only if it is a solution of Eq. (8) so that ( x* ) k r 1 X f r ( x* )

546 T.K. Roy Proof . The proof of Theorems 9 12 are followed by Miettinen (1999). When f r (x ) and (r = 1, 2,…, k) and g j (x) (j = 1, 2, …, k) are polynomial and signomial func tions, Eqs. (4), (5), and (7) may be reduced to a single objective geometric pro gramming problem. 3. FUZZY GEOMETRIC PROGRAMMING TECHNIQUE TO SOLVE A MULTI-OBJECTIVE GEOMETRIC PROGR AMMING PROBLEM Multi-objective geometric programming (MOGP) is a special type of a class of MON LP problems. Biswal (1992) and Verma (1990) developed a fuzzy geometric programm ing technique to solve a MOGP problem. Here, we have discussed a fuzzy geometric programming technique based on max min and max convex combination operators to solve a MOGP. 1, 2, , m) are When fr (x) (r 1, 2, , k) and gi (x) ( j polynomial or signomial functions, Eq. (1) may be taken as a MOGP. To solve the MOGP probl em (1), we use the Zimmerman’s (1978) technique. The procedure consists of the fol lowing steps. Step 1. Solve the MOGP as a single objective GP problem using only one objective at a time and ignoring the others. These solutions are known as i deal solutions. Repeat the process k times for k different objective functions. Let x1 , x 2 , x3 , , x k be the ideal solutions for the respective objective fu nctions, where xr r ( x1r , x2 , r , xn ) Step 2. From the ideal solutions of Step 1, determine the corresponding values f or every objective at each solution derived. With the values of all objectives a t each ideal solution, the pay-off matrix of size (k × k ) can be formulated as fo llows : f1 ( x) f 2 ( x) f k ( x)

Geometric Programming: An Inventory Model 547 x1 x2 f1* ( x1 ) f 2 ( x1 ) ........ f k ( x1 ) fk ( x2 ) ......... f k* ( x k ) f1 ( x 2 ) ...... ......... x k f1 ( x k ) f 2* ( x 2 ) ........ ......... ........ f 2 ( x k ) ........ Step 3. From the Step 2, find the desired goal Lr and worst tolerable value U r of f r ( x), r 1, 2,..., k as follows: Lr fr Ur , ( r 1,2,...,k ) where Ur Max ( f r ( x 1 ), f r ( x 2 ),..., f r ( x ( r f r ( x ( r 1 ) ),..., f r ( x k )) 1) ), f r* ( x r ), Lr = Min ( f r ( x1 ), f r ( x 2 ), f r ( x ( r 1) ), , f r ( x k )) , fr ( x(r 1) ), f r* ( x r ), Step 4. Define a fuzzy linear or non-linear membership function 1, 2, 3, k ) r ( f r ( x )) for the r-th objective function f r ( x ), ( r r( f r ( x )) 1 u r ( f r ( x )) 0 if if if f r ( x ) Lr Lr f r ( x ) U r fr ( x ) U r Here u r ( f r ( x )) is a strictly monotonic decreasing function with respect t o f r ( x ). Step 5. At this stage, either a max min operator or a max convex co mbination operator can be used to formulate the corresponding single objective o ptimization problem. 3.1 Through a Max Min Operator

According to Zimmermann (1978), the problem (1) can be solved as:

548 D T.K. Roy ( x* ) Max(Min ( 1 ( f1 ( x)), 2 ( f 2 ( x)), ..., k ( f k ( x)))) (9) subject to g j ( x) bj , j 1, 2, ,m x>0 which is equivalent to the following problem as Maximize (10) subject to r ( f r ( x)), for b j , for j r 1, 2,3, 1, 2, ,m ,k x>0 g j ( x) The parameter is called an aspiration level and represents the compromise among the objective functions. After reducing the problem (10) into a standard form of

a PGP problem, it can be solved through a GP technique. 3.2 Through a Max–Convex Combination Operator Using the membership functions r ( f r ( x )) to formulate a crisp non-linear pr ogramming model (following Tiwari et al., 1987) by adding the weighted membershi p functions together as: D( x* ) Maximize ( m r r 1 r( f r ( x ))) (11) subject to g i (x) bj , j 1, 2, , m 1 2 x > 0. m For equivalent weights, 1 are considered.

Geometric Programming: An Inventory Model 549 EXAMPLE 10. Solve MOGP Minimize Z 1 ( x ), Z 2 ( x ) (12) subject to Y ( x) where 1, x > 0 Z1(x) 2 25 x1 30 x1 2x2, Z 2 (x) T 15 x1 2 20 x11 x2 , Y (x) x11x21 and x ( x1 , x2 ) . In order to solve the problem (12), we shall have to solve the subproblems (13) Minimize Z1(x) subject to Y(x) 1 (Sub-PGP – 1 ), x >0 It is a GP with DD = 3 – (2 + 1) = 0 and Minimize subject to Y(x) 1 (Sub-PGP–2 ), x> 0 Z2(x) (14) It is a GP with DD = 3 – (2+1) = 0 Solving the sub-problems (12) and (13) by GP te chnique, we have For (Sub PGP – 1) x 1* = (1.124746, 0.8890896) and * Z 1 ( x* ) = 52.70158. For (Sub PGP – 2) x 2* = (1.414219, 0.7071040) and * Z 2 ( x* ) = 28.28 427.

550 T.K. Roy The pay-off matrix is given below: Z1 ( x ) 52.71058 60.60687 Z 2 ( x) 30.92735 28.28427 x 2 x 1 From pay-off matrix the lower and upper bounds of Z 1 ( x ) be 52.71058 and 60.6 0687 and that of Z 2 ( x ) be 30.92735 and 28.28427. [ 52.71058 Z1 ( x ) 60.60687 and 28.28427 Z2 ( x ) 30.92735 ] Suppose Z 1 ( x ) and Z 2 ( x ) are the linear membership functions of the objec tive functions Z 1 ( x ) and Z 2 ( x ) respectively and they are defined as: 1, Z1 if Z 1 ( x ) 52.71058 Z1( x ) if 52.71058 if Z 1 ( x ) Z1( x ) 60.60687 (x) 60.60687 0 1, 7.89629 60.60687 if Z 2 ( x ) 28.28427 Z2( x ) if 28.28427 if Z 2 ( x ) Z2( x ) 30.92735 Z2 (x) 30.92735 0 2.64308 30.92735 Equation (14) can be reduced to a single objective GPP by max-min operator or ma x-addition operator. 3.3 Through a Max Min Operator Using max-min operator MOGP (14) can be reduced to a following single objective problem Maximize (15)

subject to

Geometric Programming: An Inventory Model Z1 ( x) 60.60687 7.89629 Z1 ( x ) 551 30.92735 2.64308 Z 2 ( x) , Z2 ( x) , x1 1 x2 1 1, , x1 , x 2 0 , and 1 In the standard form of GP the problem (15) can be written as Minimize 1 (16) subject to 2 0.41249 x1 + 0.49499 x1 2 x 2 + 0.13029 2 0.48501 x1 + 0.64668 x 1 2 x 2 + 0.0 8546 1 1 and 1 The problem (16) has DD = 8–(3+1)=4. The corresponding dual problem (DP) is 1, , x1 , x 2 , 0 , x1 1 x 2 1 Maximize d ( w ) = 1 w1 w1 0.41249 w2 w5 B w2 A 0.49499 w3 w6 B w3 A 0.13029 w4 w7 B w4 A (17)

0.48501 w5 0.64668 w6 0.08546 w7 1 w8 subject to the following normal and orthogonal conditions are as follows: w1 1 w1 w4 w 1, 2 w2 2 w3 w5 2 w6 w8 0 w3 2 w6 w8 0

552 T.K. Roy where A w2 w3 w4, B w5 1 w6 w7 0 < w1,w 2 ,w 3 ,w 4 ,w 5 Solving the DP (17) subject to the normal and orthogonal conditions, * we get th e optimal values of dual variables w1 = 1, w* = 0.54598, w* = 3 2 * * * * 0.3061 4, w4 = 0.12419, w5 = 0.55135, w6 = 0.34351, w7 = 0.08146, and w* = 0.976311. Th e optimal dual objective value is d ( w* ) = 1.02426, 8 * and hence, the optimal values of the decision variables are x 1 = 1.16436 * * * * * and x1 = 0.85884. Then Z 1 ( x ) = 52.89796 and Z 2 ( x ) = 30.13521. 3.4 Through a Max–Convex Combination Operator Using a convex-combination operator, the multi-objective problem (12) can be tra nsformed into a following single objective problem: Maximize V ( Z1 ( x), Z2 ( x)) = 1 Z1 (x) 2 Z2 (x) =19.37661– g( x ) (18) subject to x1 1 x2 1 where 2 2 g (x) = 3.166604 x1 + 3.79925 x 1 2 x 2 + 5.67520 x1 + 7.56693 x 1 1 x 2 1 x1 , x2

0 Here 1 2 1. For maximizing the problem (18), it is sufficient to solve the following problem : Minimize g ( x ) = 2 2 3.16604 x1 +3.79925 x 1 2 x 2 +5.67520 x 1 +7.56693 x 1 1 x 2 (19)

Geometric Programming: An Inventory Model 553 subject to x1 1 x2 1 1 , x1 , x2 0 The problem has DD = 5 – (2+1) = 2. The corresponding dual problem is Maximize d ( w ) = 3.16604 w1 w1 3.79925 w2 w2 5.67520 w3 w3 7.56693 w4 w4 1 w5 (20) subject to the normal and orthogonal conditions w1 w2 w2 2 w4 w3 w5 w4 0, 1, 2 w1 0 2 w2 w3 w4 w5 1 0 w1 , w 2 , w3 , w4 , w5 * Solving the problem (20), we ultimately get w1 = 0.28032, w* , w3 = 0.39960, w4 = 0.21309, and w5 = 0.05599. The value of d . Therefore the value of g * = 17.85902 and the value of * * x * = 0.795245, and the value of objectives are Z 1 ( x * ) = 2 Z 2 ( x ) = 28.92060. 4. MULTI-OBJECTIVE MARKETING PLANNING INVENTORY PROBLEM In most inventory problems, the unit price of an item in considered as independe nt in nature. Actually, it relates to the demand of that item. When the demand o f an item is high, it is produced in large numbers. Fixed costs of production ar e spread over a large number of items. Hence the unit cost of the item decreases ; i.e., the unit price of an item inversely relates to the demand of that item. = ( 1 * 2 * * * 0.10699 w* ) = 17.85902 = 1.25747 and x * 54.61878 and

Cheng (1989) formulated the EOQ problem with this idea and solved it through the GP method. Similarly the marketing cost, which includes the advertisement and p romotion cost, directly affects the demand of an item. The manufacturing compani es increase the advertisement cost and give some advantages (like promotion, inc entives) to their sales representatives according to their performances. Lee and Kim (1993) studied the marketing planning problem considering such how to solve the problem by the GP method.

554 T.K. Roy The following basic assumptions are used in the proposed model: ASSUMPTIONS. 1. Production is instantaneous. 2. Demand is uniform. 3. The demand of a function i s directly proportional to the marketing expenditure; i.e., Di d 2 i M i i , d 2 i 0 , i 0 , 4. The unit cost is inversely proportional to demand; i.e., c0 i c 0 i Di Ti , c 0 i 0 . Let for the amount of stock is Ri at time t = 0. In the in terval ( 0 ,Ti t1i t 2 i ), the inventory level gradually decreases to meet dema nds. By this process, the inventory level reaches zero level at time t1i and the n shortages are allowed to occur in the interval ( t1i ,Ti ). The cycle then rep eats itself. The differential equation for the instantaneous inventory qi ( t ) at time t in (0, Ti) is given by dq i ( t ) = dt Di for 0 t Ti (21) with the initial conditions qi ( 0 ) Ri , q i ( Ti ) S i and q i ( t 1i ) 0. For each period, a fixed amount of shortage is allowed and there is a penalty co st c 2 i per items of unsatisfied demand per unit time. From Eq. (21) qi ( t ) Ri Di t for 0 t Ti t t 1i = Di ( t 1i t ) for t 1i So, D i t 1i Ri , S i Di t 2 i , Qi D i Ti hi c 0 i ( Q i 2Qi c 2i S i 2Qi 2 Holding cost = hi c 0 i Shortage cost = c 2 i t 1i 0 Ti ( t 1i q i ( t )dt q i ( t )) dt Si ) 2 Ti Ti

Geometric Programming: An Inventory Model 555 Qi Production cost = c 0 i Qi c 0 i d 2 iri M i ri i Advertisement cost = M i Qi The total inventory cost = setup cost + holding cost + shortage cost = hi c 0 i ( Qi S i )2 Ti 2 Qi c 3i c2i Si Ti 2 Qi 2 The total average inventory cost, T C1 ( M , Q , S ) = n i 1 hi ( Q i S i ) 2 c0 i d 2 iTi M 2 Qi Ti i c 3i d 2i M i i Qi c 2i Si 2 Qi 2 (22) = n i 1 1 hi c0 i d 2 iTi M i Ti i Qi hi c0 i d 2 iTi M i Ti i S i 2 T T 2 hi c0 i d 2 i i M i i i S i2 d 2i M i i Si c 3i c 2i 2 Qi Qi 2 Qi n And total additional cost = marketing cost + production cost = i 1 M i Qi c i Qi . So, the total average additional cost TC 2 ( M ) n i 1 d 2i M i i 1

1 d 2 i Ti c 0 i M i i ( 1 ri ) (23) Special Case. When shortages are not allowed i.e., when c 2 i TC1 ( M ,Q ) = n i 1 , then . (24) 1 hi c0 i d 2 iTi M i Ti i Qi 2 c 3i d 2 i M i i Qi T C1 ( M ) = n i 1 d 2i M i i 1 1 d 2 i ri c 0 i M i i ( 1 ri ) . (25)

556 T.K. Roy 4.1 Problem Formulation The manufacturing organization produces some items and stocks these items in a w arehouse. The manufacturing companies or organizations use a huge advertisement for their products in order to increase the level of demand. Still they have som e limitations regarding total space capacity, total allowable shortage cost, etc . In this phenomenon, the organization is interested in minimizing the inventory -related cost (including setup cost, shortage cost) and additional cost (includi ng marketing cost and production cost) simultaneously. The problem is to minimiz e total average inventory costs and also to minimize total average of additional cost under the limitations of space capacity, total allowable shortage cost. He nce the problem is Minimize TC 1 ( M , Q , S ),TC 2 ( M ) subject to n i 1 (26) Wi ( Qi Si ) W , i n c 2i S i 1 2 Qi 2 S Mi, Qi, Si, > 0 for i = 1,2,…,n. 4.2 Solution Procedure of Multi-objective Inventory Model (MOIM) The MOIM may be solved by several techniques. Some of those are the fuzzy geomet ric programming technique and global criterion method. Here global criterion is used to find the compromise solution of model (26). In this method, the objectiv e functions are combined to a single objective function. 4.2.1 Global Criterion Method Let wr 0, r 1, 2 be the normalized weights (i.e., w1 w2 1 ) corresponding to the objective functions TC 1 ( M , Q , S ) and TC 2 ( M ) . TC 01 and TC02 are the ideal objective values of TC 1 ( M , Q , S ) and TC 2 ( M ) , respectively. Dedu ctions are shown in Appendix A. TC01 and TC02 are obtained objective functions f or TC 1 ( M , Q , S ) and TC 2 ( M ), respectively,

Geometric Programming: An Inventory Model 557 without constraints by GP methods. The weighted Lp-problem according to Miettine n (1999) is 2 1/ p Minimize U p ( M , Q, S ) r 1 wr TC0 r 1/ p 1 p = w1 w2 (27) subject to same constraints as in Eq. (26) CASE 1. The weighted sum problem (i.e ., for p = 1 in (27) is given as Minimize U 1 ( M , Q , S ) w1 ( TC1 ( M , Q , S ) TC01 ) w2 ( TC2 M TC02 )) subject to same constraints as in Eq. (26). Since w1 , w2 ,TC 01 and TC 02 are i ndependent of the decision variable, so it is enough to solve the following prob lem: Minimize V1 ( M , Q , S ) w1TC 1 ( M , Q , S ) w 2 TC 2 M (28) subject to same constraints as in (26) where U 1 ( M , Q , S ) V1 ( M , Q , S ) ( w1TC 01 w2 TC 02 ) . The problem is a signomial GP problem with DD = 6n – 1 and can be solved by the GP method. CASE 2. The least squares problem (i.e., for p = 2 ) is given as Minimize U 2 ( M , Q, S ) w1 (TC1 ( M , Q, S ) TC01 ) 2 w2 (TC2 M TC02 ) 2 1/ 2 subject to same constraints as in Eq. (26). TC1 ( M , Q, S ) TC01 TC2 ( M ) TC02 p p [TCr . p

558 T.K. Roy To obtain the standard form of a GP problem of the above weighted quadratic prob lem, we introduce two new variables y 1 and y 2 , which are the upper bounds of TC 1 (.) TC 01 and TC 2 (.) TC 02 respectively ( i .e ., TC 1 ( M , Q , S ) TC 0 1 y 1 and TC 2 M TC 02 y 2 ). We may then rewrite the problem as: 2 Minimize V 2 ( M , Q , S ,Y ) w1 y 1 2 w2 y 2 (29) subject to TC 1 ( M , Q , S ) TC 01 1 w n i 1 y1 TC 01 1, TC 2 ( M ) TC 02 1 s i n i 1 y2 TC 02 2 1 W i Qi Si ) 1 , c 2i S i 2Qi 1 M i ,Qi , S i , y 1 , y 2 0 for 1, 2 ,.., j where U 2 ( M , Q, S ) (V2 ( M , Q, S ))1/ 2 The problem (29) is also a signomial GP problem with DD = 6n + 1 and it can be s olved by the GP method. CASE 3. The Tchebycheff problem (i.e., for p Minimize Maximize M ,Q , S r 1, 2 wr TCr . ) is given as TC0 r subject to same constraints as in Eq. (26) . We introduce a new variable w1 ( TC 1 ( M , Q , S ) TC 01 ) and ( i.e., w1 ( TC 1 ( M , Q , S ) TC 01 ) w 2 ( TC 2 ( M ) TC 02 ) ).

, which is maximum between w 2 ( TC 2 ( M ) TC 02 ) and

Geometric Programming: An Inventory Model 559 The problem is then reorganized as Minimize subject to TC 1 ( M , Q , S ) TC 01 1 w n i 1 (30) w1TC 01 1, TC 2 ( M ) TC 02 1 s n i 1 w 2 TC 02 2 1 Wi ( Qi Si ) 1 , c 2i S i 2 Qi 1 y1 , y2 , M i , Qi , Si 0, i 1, 2, ,n The problem is again a signomial GP problem with DD = 6n + 1 and it can be solve d by the GP technique. 4.3 Numerical Illustration A multinational soft drink manufacturing company produces two types of brands. T he brands are produced in lots. The pertinent data for the items are given in Ta ble 1. Table 1. Input Data for Model 17 Brands names Inventory holding cost rate (hi) S hortage cost (c2i) ($) Set up cost (c2i) ($) Annual demand (Di) Production cost (c0i) ($) Storage area (wi) (m2) A 25% 10 130 B 32% 14 150 1. 2 2 4.34 M 2 1.464 10 M 1 1.5 12 M 5.05 M 1 1.8

3 2.5 Total available storage area and total allowable shortage cost are w = 225 m2 an d S = $ 0.085

560 T.K. Roy The ideal value (as computed in Appendix A) of TC1(M,Q,S) is TC01 = $123.1274 an d that of TC2(M) is TC02 = $122.2257. The company decides to know the optimal va lues of the inventory related cost (TC1(M,Q,S)), additional cost (TC2(M)), marke ting cost M1,M2, lot sizes Q1,Q2 , shortage amount S1, S2. Optimal solutions of problem (26) are given in Table 2, Table 3, and Table 4 for different preference values of the objective functions. Table 2. Equal Preference Values of the Objective Functions i.e., for (w1, w2) = (0.5, 0.5) p 1 2 i 1 2 1 2 1 2 Mi * Qi* 39.51303 43.79591 37.23608 46.52790 32.01543 52.77223 Si * * TC 1 ( M * , Q * , S * ) TC * ( M * ) 2 123.0328 123.6351 125.2465 0.9300382 0.9025289 0.9196067 0.9720086 0.8458588 1.0992430 0.5809763 0.5143728 0.5836205 0.5108576 0.5797730 0.4950188 128.0386 127.6902 127.5276 The above table gives different optimal solutions when the decision maker suppli es equal preferences to the inventory-related cost function * TC1 ( M ,Q , S ) a nd additional cost function TC 2 ( M ) . TC1 ( M * ,Q* , S * ) is * * , whereas TC 2 ( M ) is minimum when p = 1. minimum when p Table 3. More Preference Values to the Inventory Related Cost Functions, i.e., f or (w1, w2) = (0.6, 0.4) p 1 2 i 1 2 1 2 1 2 M* Q* 0.9306990 38.83385 0.9257421 44.61137 0.9123236 0.9891751 0.8458587 1.099243 S* 0.5819986 0.5135849 * * TC 1 ( M * , Q * , S * ) ($) TC 2 ( M * )($) 127.8774 127.6460 127.5276 123.2292 123.7997 125.2465 36.57991 0.5838419 47.31423 0.5095112 32.01543 52.77223 0.5797731 0.4950187 Table 3 shows preference to ditional cost , Q * , S * ) different optimal solutions when the decision maker supplies more the inventory-related cost function TC 1 ( M , Q , S ) than the ad function TC 2 ( M ). Here * ), whereas TC * ( M * ) is TC 1 ( M * is minimum when ( p 2 minimum when p = 1.

Geometric Programming: An Inventory Model 561 Table 4. More preference values to the Additional Cost Functions i.e for (w1, w2 ) = (0.3, 0.7)) p 1 2 i 1 2 1 2 1 2 M* 0.8966431 0.8438916 0.9293127 0.9373176 0 .8961108 1.020650 Q* 38.19715 40.85587 S* TC 1 ( M , Q , S ) ($) TC 2 ( M )($) 0 .5651637 128.8857 122.5162 0.5020723 127.8178 127.5862 123.3277 124.1366 * * * * * * 38.46130 0.5824761 45.05847 0.5130581 35.32355 48.81861 0.5837589 0.5063539 The Table 4 shows different optimal solutions when the decision maker supplies m ore preference to the additional cost function TC 2 ( M ) than to * the inventor y-related cost function TC 1 ( M , Q , S ) . Here TC 1 is minimum * * when ( p ) , whereas TC 2 ( M ) is minimum when p = 1. 5. CONCLUSION Here we have discussed multi-objective geometric programming based on the global criterion method and then fuzzy geometric programming technique. We have also f ormulated the multi-objective inventory optimization model of the economic produ ction and the marketing planning problem. The different objective functions are combined into a single objective function by the global criterion method. The GP technique is used to derive the optimal solutions for different preferences on objective functions. In Tables 2–4 we have shown the optimal solution of our probl em for different preference values of the objective functions. This multiobjecti ve inventory model may also be solved by the fuzzy geometric programming techniq ue. REFERENCES Biswal, M.P., 1992, Fuzzy programming technique to solve multiobjective geometri c programming problems, Fuzzy Sets and Systems, 51: 67–71. Changkong, V, and Haime s Y.Y., 1983, Multiobjective Decision-Making, North-Holland Publishing, New York . Cheng, T.C.E, 1989, An economic production quantity model with demand dependen t unit cost, European Journal of Operational Research, 39: 174 – 179. Duffin, R. J ., Peterson, E. L., and Zener, C. M., 1967, Geometric Programming, John Wiley an d Sons, New York.

562 T.K. Roy Aliprantis, C.D., Tourky, Yannelis, N. C., 2001, A Theory of Value with Non-line ar Prices, Journal of Economic Theory, 100: 22 72. Ehrogott, M., 2005, Multicrit eria Optimization, Springer, Berlin. Lee, W.J., and Kim, D.S., 1993, Optimal and heuristic decision strategies for integrated production and marketing planning, Decision Sciences, 24: 1203–1213. Miettinen, K.M., 1999, Non-linear Multiobjectiv e Optimization, Kluwer’s Academic Publishing, Dordrecht. Tiwari, R.N., Dharman, S. , and Rao, J.R., 1987, Fuzzy goal programming an additive model, Fuzzy Sets and Systems, 24: 27 34. Verma, R.K., 1990, Fuzzy Geometric Programming with several objective functions, Fuzzy Sets, and Systems, 35: 115–120. Zimmermann, H.J., 1978, Fuzzy linear programming with several objective functions, Fuzzy Sets and Syste ms, 1: 46–55.

Geometric Programming: An Inventory Model 563 APPENDIX A Working rule for finding the ideal objective values TC 01 and TC 02 . Minimize TC 1i ( M i ,Q i , S i ) i 1 hi c0 i d 2 i i M 2 i i i i i Qi

hi c0 i d 2 i M

i i Si hi c0 i d 2 i i M i 2Qi c3i d 2iM Qi i i S1 2 c2i Si 2Qi 2 subject to M i , Qi , S i 0. The above problem is a primal GP problem with DD = 1 0. The corresponding dual p rogramming problem is hi c 0 i d 2 i i 2 w1i c 3i d 2 i w4 i w4 i T w 1i Maximize hi c 0 i d 2 i i 2 w3i T

dwi w3i

hi c 0 i d 2 i i w2i w5i

T

w2i c 2i 2 w5 i subject to normally and orthogonal conditions w1i w2 i w3i w4i w 5i 1 i i w3i i w4i 0 i i w1i w1i w2 i w3i i i w2 i w4i w 5i 0 0 2 w3i 2 w5i w1i , w2i , w3i , w4i ,w 5i 0 Solving the dual weights in terms of w3 i , we get w1i w3 i 2 i 1 i 2 , w2 i 2 w3 i i i 1 , w4 i 1 , w5 i 2 i 1 i 2

564 T.K. Roy Since the dual weights are always positive, i 1 . Substituting the above dual we ights into the dual function and then differentiating them with respect to w 3 i , we get w* i 3 ( i 1) 2 i 2 The other dual weights are * 1i i 2 , * 2i i 1, 1 4i 2 , * 5i i 1 2i where i 1 Substituting the dual weights into the dual function, we get dwi * . Following D uffin et al. (1967) we get the optimum objective value as * TCli dwi * . The dec ision variables can be obtain from the following relations: hi c 0 i d 2 i i M i 2 * 1i i i Qi

dw i , * c 3i d 2 i M i i * 4 i Qi dw i , * c 2i S i 2 * 5 i Qi dw i * Solving the above relations, we get 1 i( i 1) M i* hi c 0 i c 3 i 2 d 2 ii 1 * 1i * * 4 i ( dw i )2 1 i 1 Qi * c 3i d 2 i * * 4 i dwi hi c 0 i c 3 i 2 d 2 ii 1 * 1i * * 4 i ( dwi )2 1 i 1 Si * 2 * * 5 i dw i c 3i d 2 i * * 4 i dw i hi c 0 i c 3 i 2 d 2 ii 1 * 1i * 4i 0.5 c 2i ( dw i * ) 2

Geometric Programming: An Inventory Model 565 The ideal objective value TC01 is defined as TC 01 * n i 1 TC 1i In a similar way, we can find the optimal value of the objective function TC 2 i ( M ). Minimize TC2i ( M ) subject to M > 0. d 2i M i i 1 1 d 2i i c0i M i i (1 i) The above problem is a primal problem with DD = 0. The corresponding dual functi on is Maximize dwi d 2i 6i 6i 1 d2i i c0i 7i 7i subject to the normality and orthogonal conditions 6i 7i 1 ( 7i i 1) 6i i(1 i ) 7i 0

where 6i , 0. Solving the dual weights, we get * 6i i( i i i 1) , 1 * 7i i i i 1 , 1 i 1. Substituting the dual weights into the dual function, we get dwi * . Following D uffin et al. (1967), the optimal objective value is TC * i dwi * . 2 The optimal values of the decision variable are obtained from d 2i M i i 6i 1 dwi *

566 T.K. Roy Solving the relation, we get * * 6 i dw i 1 i 1 Mi * d 2i . The ideal objective TC 02 is defined as TC 02 * n i 1 TC 2 i .

FUZZY GEOMETRIC PROGRAMMING WITH NUMERICAL EXAMPLES Tapan Kumar Roy Department of Mathematics, Bengal Engineering and Science University, Shibpur, W est Bengal, India Abstract: Geometric programming (GP) has the high potential to be applied to a wide range of problems. This chapter summarizes the fundamentals of fuzzy GP and presents m any application examples. Some variants of the gravel box problem are presented to solve it by fuzzy GP. Fuzzy GP, gravel box problem, posynomial, signomial Key words: 1. 1.1 INTRODUCTION Geometric Programming Geometric programming (GP) can be considered to be an innovative modus operandi to solve a nonlinear problem in comparison with other nonlinear techniques. It w as originally developed to design engineering problems. It has become a very pop ular technique since its inception in solving non-linear problems. The concept o f geometric programming (GP) was introduced by Duffin et al. (1967) in their fam ous book Geometric Programming—Theory and Application. This publication is a landm ark in the development of GP. It studied all the theoretical developments up to date providing important examples to illustrate the technique. In addition to el egant proofs, it provided several constructive transformations and approximation for expressing optimization problems in suitable form in order to solve by GP. C. Kahraman (ed.), Fuzzy Multi-Criteria Decision Making. © Springer Science + Busi ness Media, LLC 2008 567

568 T.K. Roy The study of GP by Duffin et al. (1967) deals with the problem involving only a positive coefficient for the component cost terms. However, many real-world prob lems comprise of positive as well as negative coefficients for the cost terms. P assy and Wilde (1967) made a significant methodological development of GP to dea l with this type of problem. They extended the concept of the GP technique to ge neralized polynomials free from a restrictive environment. Now GP is capable of dealing with any problems involving signomials in both objective and constraint functions. It is important to note that any nonlinear algebraic problem can be t ransformed into an equivalent posynomial/signomial. For a detailed discussion, o ne may consult with the book Applied Geometric Programming written by Beightler and Phillips (1976). The advantages of GP are as follows: This technique provide s us with a systematic approach for solving a class of nonlinear optimization pr oblems by finding the optimal value of the objective function and then the optim al values of the design variables are derived. This method often reduces a compl ex nonlinear optimization problem to a set of simultaneous equations. This appro ach is more amenable to the digital computers. This method allows an easy sensit ivity analysis, which can be performed in the optimal solution. GP inherits some drawbacks. The main disadvantages of GP lie in the fact that it requires the ob jective functions and constraints in the form of posynomials/signomials. Note. S omeone guesses that the name GP comes from the many geometrical problems. There is a difference between GP and geometric optimization (GOP). GP is an optimizati on problem based on the arithmetic- geometric mean inequality (A.M. G.M.). Howev er, GOP is an optimization problem involving geometry. GP is an optimization pro blem of the form Minimize g 0 (t) subject to gj(t) 1, j = 1, 2,…, m k =1, 2,…, p (1) hk(t) = 1,

Fuzzy Geometric Programming Applications 569 ti > 0, i = 1, 2,…, n where gj(t) (j = 0, 1, 2, …, m) are posynomial or signomial functions, hk(t) (k =1 , 2,…, p) are monomials and t is the decision variable vector of n components ti ( i = 1, 2,.., n). The problem (1) may be written as: Minimize g 0 t subject to g j t 1 (2) j =1, 2, ... , m t > 0, [since g j t 1 , h j t 1 g j t 1 where g j t (=gj(t)/hk(t)) be a posynom ial (j =1, 2,.., m ; k = 1, 2, …, p)]. 2. 2.1 POSYNOMIAL GEOMETRIC PROGRAMMING PROBLEM Primal Problem Minimize g 0 (t) subject to gj t (3) 1, ( j = 1, 2, …, m) and ti > 0, (i = 1, 2, …, n) Nj where g j t = where cjk(> 0) and are real numbers. t ( t1, t2, …, tn)T. c jk k 1 n i 1 ti jki jki ( i = 1, 2, …, n; k = 1, 2,…, Nj; j = 0, 1, 2,…, m) It is a constrained posynomial primal geometric problem (PGP). The number of ine quality constraints in the problem (3) is m. The number of terms in each posynom ial constraint function varies, and it is denoted by Nj for each j = 0, 1, 2, …, m .

570 T.K. Roy The degree of difficulty (DD) of a GP is defined as number of terms in a PGP (nu mber of variables in PGP + 1). 2.2 Dual Problem: The dual programming of (3) is as follows: m Nj Maximize d( w ) = j 0 k 1 c jk w j0 w jk w jk (4) subject to N0 w 0k 1 (normality condition) k 1 m Nj j 0k 1 jki w jk 0 , (i = 1, 2, ..., n) (orthogonality condition) Nj wj0 = w jk k 1 0, wjk 0, (i = 1, 2,..., n; k =1, 2,…, Nj), w00 = 1. m There are n+1 independent dual constraint equalities and N = Nj j 1 independent dual variables for each term of the primal problem. In this case DD

= N (n+1). 3. 3.1 SIGNOMIAL GEOMETRIC PROGRAMMING PROBLEM Primal Problem Minimize g 0 (t) subject to gj t j (5) , (j = 1, 2, …, m)

Fuzzy Geometric Programming Applications 571 and ti > 0, where g j t = (i = 1, 2, …, n) Nj jk k 1 c jk n i 1 ti jki (j = 0, 1, 2, …, m) (j = 0, 1, 2, …, m; k = 1, 2, …, Nj), j t = 1 ( j = 2, …, m), ( t1, t2, …, tn)T. jk = 1 3.2 Dual Problem The dual problem of (5) is as follows: m Nj 0 j 0k 1 Maximize d(w) = subject to N0 0k k 1 m Nj jk j 0k 1 c jk w j0 w jk jk w jk 0 (6) w 0k 0 (normality condition) jki w jk 0 (i =1, 2, ..., n) (orthogonality condition)

where w00 = 1 0 j = 1 (j = 2, …, m), jk = 1 (j =1, 2, …, m; k = 1, 2, …, Nj), and Nj jk w jk k 1 1, 1 and non-negativity conditions, wj0 j 0, jk 0, (j =1, 2, …, m; k = 1, 2, …, Nj) and w00 = 1. 4. FUZZY GEOMETRIC PROGRAMMING (FGP) ~ Minimize g 0 (t) (7) bj (j = 1,2 ,…, m) subject to gj(t) t 0 ~

572 T.K. Roy ~ Here, the symbol “ Min i mize ” denotes a relaxed or fuzzy version of “Minimize.” Simila rly, the symbol “ ” denotes a fuzzy version of “ . ” ~ These fuzzy requirements may be quantified by eliciting membership functions j g j t (j = 0, 1, 2,…, m) from the decision maker for all functions gj(t) (j = 0, 1, 2,…, m). By taking account of the rate of increased membership satisfaction, the decision maker must determine the subjective membership function j g j t . It is , in general, a strictly monotone decreasing linear or non linear function u j g j t with respect to gj(t) (j = 0, 1, 2,…, m). Here for simplicity, linear members hip functions are considered. The linear membership functions can be represented as follows: 1, j if g j (t ) g j g j g0 j g j (t ) g j g j gj t = g j (t ) g 0 j if g 0 j if g j (t ) 0 for j = 0, 1, 2,…, m. gj t j 1 gj, 0 gj t g0 j g0 j g0 j gj t g j g j

gj t g j g j( t ) Figure 1. Membership function As shown in Figure 1, g0 j j the value of gj(t) such that the grade of membership function gj t is 1.

Fuzzy Geometric Programming Applications 573 g j j the value of gj(t) such that the grade of membership function gj t is 0. gj the intermediate value of gj(t) between g 0 and g j (i.e., g j ( g 0 , j j ,1). g j )) such that the grade of membership function

The problem (7) reduces to FGP when g0(t) and gj(t) are signomial and posynomial functions. Based on fuzzy decision making of Bellman and Zadeh (1972), we may w rite (i) D t* max min j gj t (max min operator) (8) subject to gj t = g j g j (t) (j = 0, 1, 2,…, m) j g j g0 j t>0 (ii) m D t* max j 0 j j gj t (max additive operator)

(0

(9) subject to t>0 j gj t = g j g j g j (t ) g0 j (j = 0, 1, 2,…, m) m (iii) D t* max j 0 ( j g j (t ) ) j (max product operator) (10) subject to gj t j = g j g j g j (t ) g0 j (j = 0, 1, 2,…, m) t > 0.

574 T.K. Roy Here, for j (j = 0, 1, 2, ..., m) are numerical weights considered by a decision making unit. For normalized weights m j j 0 1 and j 0 ,1 For equal importance of objective and constraint goals, j = 1. In contrast to GP , FGP in general has not been widely circulated in the literature. In 1990, Verm a studied a new concept to use the GP technique for multi-objective fuzzy decisi on-making problems. He projected the very importance on the product operator, wh ich reduces the DD with a considerable amount. Biswal (1992) applied the fuzzy p rogramming technique to solve a multi-objective GP problem as a vector minimizat ion problem. A vector maximization problem can be transformed into a vector mini mization problem. Cao (1993, 1994) discussed the properties of a kind of posynom ial GP with an L-R fuzzy coefficient in objectives and constraints. In the seque l, Cao (2002) published an important book on FGP, which was the most recent book until now. If gj(t) (j = 0,1,2,…,m) be posynomial function as gj(t)= Nj n c jk k 1 i 1 ti jki ( cjk(> 0) and jki ( i = 1, 2, ..., n; k = 1, 2,..,Nj; j = 0, 1, 2,…, m)) then i) max min operator (8) reduces to Maximize g j j gj t j g g0 j subject to r g r g r t g r g r0

g j j gj t g0 j g j , (r = 0, 1, 2, …, m and r j) t > 0.

Fuzzy Geometric Programming Applications 575 So V * t * = M g j j g j g0 j V* t * where t * is obtained by solving the following signomial GP: Nj n Minimize V(t) = subject to Nj r j g j g0 j c jk k 1 i 1 ti jki (12) n Nj n g

r

g 0 r k 1 crk i 1 r ti r rki j

g

0 j k 1 c jk i 1 ti a jki g g g r0 r g g g0 j j (r = 0, 1, 2, …, m and r t>0 j) ii) max additive operator (9) reduces to Nj m n g j j k 1

c jk g g i 1 0 j ti jki (13) a Maximize VA(t) = j 0 subject to t>0 So the optimal decision variable t* with the optimal objective va lue V*(t*) m j gj * can be obtained by V*(t*) = U * t * where t is the optimal g j g 0 j 0 j decision variable of the unconstrained geometric programming pro blem

g

j j j

j

576 m Nj j T.K. Roy n a Minimize U(t) = j 0 g

g 0 j k 1 C jk i 1 ti jki (14) subject to t > 0. iii) Similarly, Eq. (10) can be solved by GP based on a suitab le transformation. 4.1 Numerical Example 1 Minimize Z (x) x 1 1x 2 2 [Here objective goal Z(x) 6.94 with tolerance 0.19 ] ~ (15) Y1 (x ) Y 2 (x ) 2x 1 2 x 23 57.87 (with tolerance 2.88) x1 x2 1, x1, x 2 0 Here, linear membership functions for the fuzzy objective and constraint goals a re 1, 1 if Z 1 (x) 1 (x ) 6.75 6.94 x = 6.94 0.19 0 if 6.75 Z 1 (x) if Z 1 (x ) 6.94

¡

j

1, 2 if Y 1 (x ) 1 (x ) 57.87 x = 60.75 2.88 0 if 57.85 if Y 1 (x) Y 1 (x ) 60.75 60.75

Fuzzy Geometric Programming Applications 577 i) Based on max–min operator (8), FGP (15) reduces to Maximize V(x1, x2) = 6.94 x 1 1x 22 0.19 (16) subject to 60.75 x 1 1x 22 2.88 6.94 x 1 1x 22 0.19 x1+ x2 1, x1 >0, x2 > 0 So Eq. (16) reduces to Maximize VM(x1, x2) = 36.37 V(x1, x2 ) subject to 0.341 x 1 1 x2 2 0.045 x1 2 x2 3 where V(x1, x2) = 5.26 x1 1 x2 2 To solve Eq. (17), we are to solve the following crisp GP: Minimize V(x1, x2) = 5.26 x1 1x 22 . subjec t to 0.45 x1 2 x2 3 0.341 x1 1 x2 2 For this problem DD = 5 (17) 1 , x1 > 0, x2 > 0 1 , x1+ x2 1, x1 > 0, x2 > 0. (2 + 1) = 2.

578 T.K. Roy The dual problem (DP) of this GP is Maximize d ( w01 , w11 , w12 , w21 , w22 , 5.26 w01 0 1 w22 w22 w11 w12 w01 0.045 w11 w11 0.341 w12 w11 w12 w21 0 )= w12 1 w21 w22 w21 w22 w21 0 subject to 0 w01 1 w21 0, 2 w01 3w11 2w12 w22 0 w01 2w11 w12 Considering 0 1 , we have w01 1 , w21 Here 1 11 2 w11 w12 1 , w22 w11 12 3w11 2 w12 2 w12 1 = w11 w11 w12 0.341 w12 2 w11 0. 1 w12 1 2 w11 w12 1 So, max d w11 , w12 = 5.26 0.045 w11 1 w10 3w11 2 w12 2 2 5 w11 3 w12 3 3w11 w11 w12 2 w12

w12 w11 5w11 3w12 3 For optimality of d w11 , w12 , we have d w11 , w12 w11 and 0

Fuzzy Geometric Programming Applications 579 d w11 , w12 w12 That is, 0.045 w12 w11 0. 5w11 3w12 3 = w11 2 w11 w12 1 5 2 3w11 2 w12 2 3 and w12 (2 w11 w12 1)(3w11 2 w12 2) = 0.341( w12 w11 )(5w11 3w12 3)3 . the optimal solution is d ( w) 35.75646422 , w01 1 , w11 0.2901869 , w12 0.51038 87 , w21 1.0699851 , w22 1.8497833 , 1 0.2202018 . So, the optimal solution of E q. (13) is x1 = 0.36631095, x2 = 0.633699788, and Z(x) = 6.79798859. ii) Based o n the max additive operator (9), FGP (15) reduces to Maximize VA(x1,x2)=( subjec t to x1+ x2 1, x1 > 0, x2 > 0 6.94 x1 1 x2 2 0.19 + 60.75 x1 1 x2 2 2.88 ) 57.62 V ( x1 ,x2 ) (18) where V ( x1 , x2 ) 5.263x1 1 x2 2 0.694 x1 2 x2 3 . To solve Eq. (18), we are to solve the following crisp GP: Minimize V (x1 , x 2 ) subject to x1+ x2 1, x1 > 0, x2 > 0. 5.263 x1 1 x2 2 0.694 x1 2 x2 3 For this problem, DD = 4 (2 + 1) = 1

580 T.K. Roy The DP of this GP is Maximize 5.263 = w01 subject to w01 d ( w01 , w02 , w11 , w12 ) 0.694 w02 w02 1 w11 w11 1 w12 w12 w11 w12 w11 w12 w01 + w02 =1, 2 w01 3 w02 So, w02 w01 2 w02 w12 0 2 w01 , w12 5.263 w01 w01 0.694 1 w01 w11 0 1 w01 , w11 d ( w01 ) 3 w01 1 w01 1 2 w01 2 w01 Maximize 1 3 w01 3 w01 5 2 w01 5 2 w01 subject to 0 < w01 < 1 For optimality of d ( w01 , w02 , w11 , w12 ) , we have d d ( w01 ) dw01 0 That is, 5.263(1 w01 ) (2 w01 ) (3 w01 ) The optimal solution is d ( w01 ) = 56. 10412298 w01 0.6375822, w02 0.3624178, w11 0.694 w01 (5 2 w01 ) 2 . 1.3624178, and w12

2.3624178 So, the optimal solution of Eq. (15) is x1 = 0.364517711, x2 = 0.635681102, and Z(x) = 6.788952396

Fuzzy Geometric Programming Applications 581 iii) Based on the max product operator (10), FGP (15) reduces to Maximize Vp(x1, x2)=( subject to x1+ x2 1, x1 > 0, x2 > 0 60.75 x1 1 x2 2 6.94 x1 1 x2 2 )= 770.479 V ( x1 , x2 ) (19) ) ( 2.88 0.19 where V ( x1 , x2 ) 111.018 x1 1 x2 2 25.349 x1 2 x2 3 3.653x1 3 x2 5 . To solve Eq. (19), we are to solve the following crisp GP: Minimize V ( x1 , x2 ) 111.018 x1 1 x2 2 subject to x1+ x2 1, x1 > 0, x2 > 0 25.3492 x1 2 x2 3 3.653x1 3 x2 5 For this case DD = 5 (2 + 1) = 2. The DP of this GP is Maximize 111.018 w01 w12 w11 w01 d ( w01 , w02 , w11 , w12 ) 25.349 w02 w12 w11 w12 w02 3.653 w03 w03 1 w11 w11 1 w12 subject to w01 w02 w03 1 w11 0 w01 2w02 3w03

582 T.K. Roy 2 w01 3w02 5w03 So, w03 w12 0 w01 w02 1 w11 w12 = 3 2 w01 w02 5 3w01 2w02 For the optimality of d ( w01 , w02 , w11 , w12 ) , we have w01 d ( w01 , w02 , w11 , w12 ) 0, w02 d ( w01 , w02 , w11 , w12 ) 0. That is 111.018 (3 2 w01 w02 ) 2 (5 3w01 2w02 )3 ( w01 w021 ) = 3.653w01 (8 5w01 3w02 )5 and 25.349 (3 2 w01 w02 ) (5 3w01 2 w02 ) 2 ( w01 w02 1) = 3.653w02 (8 5w01 = 769.8551092 , 3w02 )3 . The optimal solution is d( w ) w01 0.9774833, w02 0.9249442, 0.2176617 w03 0.9024275 , w11 0.1200892 , and w12 So, the optimal solution of Eq. (15) is x 1 0.394682105 , x2

0.611383637 , and Z (x ) 6.778364884 .

Fuzzy Geometric Programming Applications 583 5. 5.1 APPLICATION Gravel Box Problem Problem 1a. A total of 80 cubic-meters of gravel is to be ferried across a river on a barge. A box (with an open top) is to be built for this purpose. After the entire gravel has been ferried, the box is to be discarded. The transport cost per round trip of barge of box is Rs 1; the cost of materials of the sides and b ottom of the box are Rs 10/m2 and Rs 80/m2 and the ends of box are Rs 20/m2. Fin d the dimension of the box that is to be built for this purpose and the total op timal cost (see Figure 2). Figure 2. Gravel box problem Let us assume the gravel box has length = t1m , width = t2 m , height = t3 m The area of the ends of the gravel box = t2t3 m 2 The area of the sides of the g ravel box = t1t3 m 2

584 T.K. Roy The area of the bottom of the gravel box = t1t2 m 2 The volume of the gravel box = t1t2t3 m3 Cost function Transport cost: Rs 1/ trip 80m3 = Rs. 80t1 1t2 1t3 1 , t1t2t3 m 3 / trip Material cost: Ends of box: 2 Rs 20 / m 2 t2t3 m 2 = Rs. 40t 2 t 3 Sides of box : 2 Rs 10 / m 2 t1t3 m 2 = Rs. 20t1t3 Bottom of box: Rs 80 / m 2 t1t2 m 2 = Rs.8 0 t1t2 The total cost (Rupees): g t 80t1 1t2 1t3 1 40t2 t3 20t1 t3 80t1 t2 It is a posynomial function. As stated, this problem can be formulated as an unc onstrained GP problem Minimize g t 80 t1 1t2 1t3 1 40t2 t3 20t1 t3 80t1 t2 subject to t1, t2, t3 0 The optimal dimensions of the box are t1*=1m, t2*=1/ m, and t3*=2m, and the minimum total cost of this problem is Rs 200. Problem 1b. We now consider the following variant of the above problem (a similar discussion t ake place in Duffin et al., 1967 in their book). It is required that the sides a nd bottom of the box should be made from scrap material, but only 4 m2 of this s crap material are available. This variation of the problem leads us to the follo wing constrained posynomial GP problem:

Fuzzy Geometric Programming Applications 585 Minimize g 0 t subject to g1 t where t1 0, t2 80 t1t2t3 40t2t3 4 2t1t3 t1t2 0, t3 0 Solving this constrained GP problem, we have the minimum total cost Rs 95.24, an d the optimal dimensions of the box are t1*=1.58m, t2*=1.25m, and t3*=0.63 m. Pr oblem 1c. We now consider the fuzzy objective and constraint goal in Problem 1b. The fuzzy problem becomes Find t = ( t1, t2, t3)T so as to satisfy g 0 (t) ~ 90 and g 1 (t) ~ 4, t>0 For treating the above fuzzy inequalities, we construct the following linear mem bership functions: 0, g1 if g1 (t1 , t2 , t3 ) 4 2 , if 4 6 6 t1 , t2 , t3 = 1 g1 (t1 , t2 , t3 ) 1, g1 (t1 , t2 , t3 ) 4 if g1 (t1 , t2 , t3 ) 1, g0 if g 0 (t ) 8 if 90 90 g 0 (t ) 98 4 6 6 t = 98-g 0 (t ) , 0, 1, 98 if g 0 (t ) if g1 (t ) 2

g1 t = 6-g 1 (t ) , 0, if 4 g1 (t ) if g1 (t )

586 T.K. Roy where 8 (= 98 90) and 2 (= 6 4) are subjectively chosen constants expressing the limit of the admissible violations of the inequalities. It is assumed that the membership function g 0 t should be 1 if the objective goal is well satisfied, a nd 0 if the objective goal is violated beyond its limit 8(= 98 90) and linear fr om 0 to 1. Following the fuzzy decision on the max additive operator (9), the sa id problem can be transformed into the following equivalent conventional nonline ar programming problem as Maximize V(t) = 98 g 0 t 6 g1 t + 8 2 subject to t > 0. So the optimal decision variable t* can be obtained by solving the following unconstrained GP problems: Minimize U(t) = subject to t > 0. Mini mize U t g0 t g t + 1 8 2 10t1 1t2 1t3 1 5t2 t3 t1 t3 1 t1 t2 2 subject to t1, t2, t3 > 0. * * * Solving the unconstrained GP we have t1 = 1.58, t 2 = 1.426883, and t 3 = 0.7134413; the optimal objective goal is * * 90.45766 and the constraint goal g 1* t * = 2t1 t3 * g0 t* = 80 ttt * * * 1 2 3 * * 40t2t3 = * t1*t2 = 4.50895.

Fuzzy Geometric Programming Applications 587 ACKNOWLEDGMENTS The author wants to thank Surapati Pramanik for his constructive comments in pre paring the manuscript. REFERENCES Beightler, G.S., and Phillips, D.T., 1976, Applied Geometric Programming, Wiley, New York. Bellman, R., and Zadeh, L.A., 1972, Decision-making in a fuzzy enviro nment, Management Sciences, 17: 141–164. Biswal, M.P., 1992, Fuzzy programming tec hnique to solve multi-objective geometric programming problems, Fuzzy Sets and S ystems, 51: 67–71. Cao, B.Y., 2002, Fuzzy Geometric Programming, Kluwer Academic P ublishers, The Netherlands. Cao, B.Y., 1993, Fuzzy geometric programming (I), Fu zzy Sets and Systems, 53: 135–153. Cao, B.Y., 1994, Posynomial geometric programmi ng with L-R fuzzy coefficients, Fuzzy Sets and Systems, 64: 267–276. Duffin, R.J., Peterson, E.L., and Zener, C., 1967, Geometric Programming Theory and Applicati ons, Wiley, New York. Passy, U., and Wilde, D.J., 1967, Generalized polynomial o ptimization, SIAM Journal on Applied Mathematics, 15(5): 1344–1356. Verma, R.K., 1 990, Fuzzy geometric programming with several objective functions, Fuzzy Sets an d Systems, 35: 115–120.

INDEX A Additive weighting 6, 7, 187 Advanced manufacturing systems 200, 215 Analytic hi erarchy process 7, 53, 55, 85, 87, 91, 119 Analytic network process 9, 209 ANFIS 304, 316 Ant colony optimization 27, 41 Approximation algorithm 325, 332, 508 A rtificial intelligence 19, 26 Artificial neural networks 26 Attainment problem 4 36, 437, 439 Auxiliary variable 24, 329, 382 Axiomatic design 209, 210 Decision support system 29, 320, 520 Decision tree models 288 Degree of satisfaction 12, 78, 247 Descriptive analysis 237 Design range 211, 229 Difference measures 171 D iscordance index 124 Disjunctive 4 Distance from target 7 Distillation chain 126 Dominance 3, 8 Dual problem 551 Dynamic programming 10, 410, 413 E Economical attributes 179 E-government 85, 86, 87 Eigenvalue technique 92 ELECTR E III 119, 120, 123 Elimination by aspects 5 Entropy value 53, 79 Environmental engineering 453, 480 EOQ problem 553 E-transformation 88, 108 Euclidean distance 534 Evolutionary algorithm 40, 523, 524 Expectation optimization model 375, 379 Expert system 14, 27, 28 Extent analysis 53, 94, 105 B Black system 456 C Capital investment 6 Common range 212 Compensatory 3, 5 Complete optimal solutio n 380, 487 Compromise approach 15, 330 Concordance index 124, 125 Conjunctive 4 Consistency ratio 92 Constrained optimization 32, 36, 524, 529 Contrary index 28 5 Convex 26, 71, 78, 134 F Feasible region 281, 399 Flexible manufacturing 42, 216, 264 Fractile criterion model 379, 396 Functional requirements 210 Fuzziness patterns 247, 258 Fuzzy con version scale 72 D Data envelopment analysis 13, 162 Data mining 281, 290 Decision making 1, 2, 9, 16, 19, 24, 166 Decision matrix 249, 285 589

590 Fuzzy geometric programming 539, 546 Fuzzy if-then rules 302 Fuzzy inference sys tem 302 Fuzzy multi-criteria decision making 263 Fuzzy optimization 455, 459 Fuz zy Sensitivity 523, 524 Index Inventory model 561 Investment costs 180 Iterative goal programming approach 431 J Judgement matrix 78 G Gaussian random variable 397 Genetic algorithm 28, 36 Geometric programming 567 Global criterion 556 Global priority 249, 263 Goal based interaction 508 Goal fu lfillment level 455 Goal programming 10, 21, 29, 283, 432 Gomory’s cutting-plane m ethod 446 Gravel box problem 583 Grey fuzzy 453 Grey number 454 Grey parameters 453 Grey related analysis 281, 283 Grey systems theory 456 K Kuhn-tucker necessity theorem 394 L Lagrange function 393 Level of satisfaction 258 Lexicographic 4 Lexicographic go al programming 432 Lexicographic semi-order 5 Linear assignment 5 Linear convex combination 78 Linear programming 327 Linguistic terms 485, 505 Locally Pareto o ptimal solution 543 Logarithmic least square 56 Logistic function 252 L-R type t rapezoidal fuzzy number 200 H Hierarchical TOPSIS 172 Hierarchy 7, 239 Hybrid method 544 M Mapping point 352 Max-additive operator 575 Maximax 4 Maximin 4 Max–min operator 3 26 Max–product operator 581 Monte Carlo simulation 281 M-pareto optimal solution 3 81 Multi-attribute 3 Multi-criteria 10 Multi-criteria decision aid 119 Multi-obj ective 10 Multi-objective linear programming 325 Multi-objective optimization 45 3 Multiplicative weighting 187 I Ideal objective value 556 Inconsistency 92 Independence axiom 210 Index of optim ism 78 Indifference threshold 123 Information axiom 209 Intangible factors 266, 268 Integer multicriteria decision-making 433 Intelligent fuzzy MCDM 263 Intelli gent optimization 26 Intelligent techniques 45 Interactive multi-objective decis ion making 39, 483 Interactive programming 375 Interactive 376 Interval numbers 281, 457 Inventory 561 N Negative ideal solution 7, 165 Neuro-fuzzy 258 Nondominated solution 532 Noncomp ensatory 3, 5

Index Nonconcave 342 Non-pareto techniques 37 Normality condition 570 Normalized fuzzy weights 197 Normative analysis 237 NSGA-II 523 Shannon entropy 212 Signomial 56 7 Signomial GP problem 557 Simple Additive Weighting 187 Simplex algorithm 335 S imulated annealing 35 Simulation 281 Stochastic programming 375 Strategic planni ng 86, 90 Strengths 91, 95 Strict nondomination 145 Strict preorder 147 Subjecti ve factor measures 263 SWOT 85 System range 211, 215 591 O Operating costs 162, 180 Opportunities 85, 95 ORESTE 8 Orthogonal conditions 551 , 565 Outranking method 8, 119 P Pairwise comparison 8, 12 Pairwise comparison matrix 54, 65 Pareto based techniq ues 38 Pareto optimal 37, 381, 392 Pareto optimality 409, 541 Pareto optimal sol ution 542, 543 Particle swarm optimization 27, 42 Positive ideal solution 165, 1 70 Positive index 285 Posynomial 539 Posynomial function 539 Preference threshol d 123 Primal problem 565 Probability maximization model 375 PROMETHEE 8, 119 T Tabu search 34 Tchebycheff problem 545 Technical attributes 163 Threats 85 TOPSI S 159 Trapezoidal fuzzy number 176 Triangular fuzzy number 190 U Utility models 8, 209 Utopia maximum 532 Utopia minimum 532 V Vector maximization problem 574 Vector minimization problem 574 Vector normaliza tion 165 Q QFD 301 Quasi-concave 339 R Ranking method 5, 145 Real fuzzy number 411 Research directions 44, 45 Robotic s ystems 159 W Waste load allocation 453 Weak domination 143 Weak preorder 146 Weakly Pareto op timal solution 543 Weaknesses 85, 96 Weighted normalized decision matrix 169 Wei ghted product 6 Well-structured problem 25 White system 456 -Pareto optimality 4 09 S Scoring methods 187 S-curve 245, 251 Semi-ill structured problems 25 Sensitivity analysis 28, 103 Seperation measures 171